MIT OpenCourseWare http://ocw.mit.edu Continuum Electromechanics For any use or distribution of this textbook, please cite as follows: Melcher, James R. Continuum Electromechanics. Cambridge, MA: MIT Press, 1981. Copyright Massachusetts Institute of Technology. ISBN: 9780262131650. Also available online from MIT OpenCourseWare at http://ocw.mit.edu (accessed MM DD, YYYY) under Creative Commons license Attribution-NonCommercial-Share Alike. For more information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.

11

Streaming Interactions

- - - - --

L~5~7

11.1 Introduction

Some of the most significant interactions between continua having large relative velocities in-volve charged particle beams accelerated under near-vacuum conditions. Thus, the first part of thischapter gives some background in electron beam dynamics. The charged particles of Chap. 5 become a con-tinuum in their own right because their inertia is dominant. Section 11.2, on the laws and theorems fora charged particle gas, draws on the fluid mechanics of Chap. 7, and leads to the steady electron flowsconsidered in Secs. 11.3 and 11.4. Flows illustrated in these latter sections are typical of thosefound in magnetrons and in electric and magnetic electron beam lenses. Pictured as they are inLagrangian coordinates, the motions appear to be time varying. But, if viewed in Eulerian coordinates,the electron flows of these sections are steady and might be considered in Chap. 9. The remaining sec-tions relate not only to electron beams, but to electromechanical continua introduced in previouschapters.

Sections 11.6-11.10 have as a common theme the use of the method of characteristics to understanddynamics in "real" space and time. The approach is restricted to two dimensions, here one space andthe other time, but makes it possible to investigate such nonlinear phenomena as shock formation andnonlinear space-charge oscillations. Thus it is that these sections are concerned with quasi-one-dimensional models. As pointed out in Sec. 4.12, the small-amplitude limits of these models are identi-cal with the long-wave limits of two- and three-dimensional models. Thus, the quasi-one-dimensionalmodel represents what physical content there is to the dominant modes from the infinite number ofspatial modes of a linear system. However, nonlinear phenomena can be incorporated into the quasi-one-dimensional model.

In addition to giving the opportunity to develop nonlinear phenomena, the method of characteristicsgives the opportunity to explore the implications of causality for longitudinal boundary conditions andthe general domain of dependence of a response pictured in the z-t plane. This gives an alternative tothe complex-wave point of view, taken up in the remainder of the chapter, in appreciating the differencebetween absolute and convective instabilities and between evanescent and amplifying waves. The proto-type configurations examined in Sec. 11.10 are analogous to traveling-wave electron beam or beam plasmasystems taken up in later sections.

Sections 11.11-11.17 return to a theme of complex waves. Spatial transients in the sinusoidalsteady state are considered in Sec. 5.17 with the tacit assumption that the response decays away fromthe excitation source. As illustrated in these sections, the response could just as well amplify fromthe region of excitation. How is an evanescent wave, which simply decays from the region of excitation,to be distinguished from one that amplifies? Temporal transients are first introduced in Sec. 5.15,and instability, defined as an unbounded response in time, illustrated in Sec. 8.9. In a system thatis infinitely long in the longitudinal direction, a dispersion relation that gives "unstable" w's forreal k's can either imply that the response is unbounded in time at a given fixed location, or thatthere is unlimited growth for an observer moving with the response. In a given situation, how is anabsolute instability to be distinguished from one that is convective? For special hyperbolic systems,these questions are answered in terms of the method of characteristics in Sec. 11.10. Sections 11.11and 11.12 are devoted to the alternative of answering these questions in terms of complex waves. Theremaining sections illustrate with classic examples.

BALLISTIC CONTINUA

11.2 Charged Particles in Vacuum; Electron Beams

Equations of Motion: In terms of the Eulerian coordinates of Sec. 2.4, Newton's law for a par-ticle having mass m and charge q, subject to the Lorentz force (Eq. 3.2.1), is

m(- + v-Vv) = q(E + vx -po ) (1)

Multiplied by the particle number density, n, this expression is almost what would be written todescribe a fluid. The pressure and viscous stress terms are absent from Eq. 1. To each point inspace is ascribed the velocity, v, of the particle that happens to be at that point at the giveninstant in time.

Because the pressure and viscous stresses are absent, much of the literature of electron beamspictures the motions in Lagrangian terms, as discussed in Sec. 2.4. Then, the initial coordinates ofeach particle are the independent variables as the partial derivative with respect to time is taken:

m = q(E + v x ý H ) (2)at0

A.t11.1 Secs. 11.1 & 11.2

Thus, for example, in cylindrical coordinates the equations of motion for a particle having the instan-taneous position (r,e,z) are

d2r de 2 dE+ Hrd (3)dt2 dt m r m oz dt

2de dr d8 1 d 2 de dz drd"2r 2+ 2 ( -- = (dHr oH - (4)dt2 dt dt r dt ordt oz dt

2d z qE - 1 HH r d (5)dt2 m z m or dt

where the second terms on the left in Eqs. 3 and 4 are respectively the centripetal and Coriolisaccelerations of rigid-body mechanics.

The dynamics of interest can be pictured as EQS with an imposed magnetic field. In Sec. 3.4 itis argued that in EQS systems, the magnetic force is negligible compared to the electric force. Now,the particles of interest include electrons or ions in vacuum. Their velocities can easily be largeenough to make magnetic forces due to the imposed field important. The arguments of Sec. 3.4 show thatthe part of the force attributable to a magnetic field induced by the displacement current (or the cur-rent density associated with the accumulation of net charge) is still negligible provided that times ofinterest are long compared to the transit time of an electromagnetic wave.

The laws required to complete the description are usually written in Eulerian coordinates, muchas in the description of charged migrating and diffusing particles in Sec. 5.2. With the charge den-sity defined as nq, conservation of charge, Gauss' law and the condition that the electric field beirrotational are written as

- + Vp = 0 (6)

V.E E = p (7)o

E = -VO (8)

Either Eq. 1 or 2 and these three expressions comprise two vector and two scalar equations in thedependent variables -, - and p,O.

Energy Equation: The equivalent of Bernoulli's equation for charged ballistic particles is ob-tained following the same steps as in Sec. 7.8. With the use of a vector identity* and Eq. 8, Eq. 1becomes

av 14 + + +mt + x v) + V( mvv + q) = q x iH0 (9)

-• 4

where the vorticity, W V x v. Because both the vorticity and magnetic field terms in this expressionare perpendicular to V, it can be integrated along a stream line joining points a and b to obtain

bS 8 41 . +4 bm *d + + q =]a0 (10)

a

In the steady state, the sum of the kinetic and electric potential energies of a particle are constant,regardless of the imposed magnetic field. Note that, in Eulerian coordinates, particle motions aresteady provided 0 is constant.

Theorems of Kelvin and Busch: The curl of Eq. 9 describes the vorticity in Eulerian coordinates:

~+ Vx ( x v) = V x (v x H) (11)at m

This expression can be integrated over an open surface S enclosed by a contour C moving with the par-ticles. The generalized Leibnitz rule, Eq. 2.6.4, then gives

*÷++ 1 -.)v.Vv = (V x v) xv + V(v.Vv 2

Sec. 11.2 11.2

adda = (x xoH).dl (12)dt S m C o

Provided there is no imposed magnetic field, the vorticity is conserved over a surface of fixed identity.The magnetic field generates vorticity.

Kelvin's theorem, represented in Eulerian terms by Eq. 12, is often exploited in Lagrangian termsin dealing with axisymmetric electron beams having no 0 components of electric or magnetic field.Then, Eq. 4 describes the 0-directed particle motions. Because particle current contributions to A areignored, a is solenoidal and can be represented in terms of a vector potential, T = [A(r,z)/r]i 0(Eq. (g) of Table 2.18.1). Thus,

1 d 2 dO = 1[ A dz 11A dr(r ) = - [ + (13)r dt dt m r 3z dt r - r dt

What is on the right in Eq. 13 is the rate of change of A(r,z) for a given particle,

d(r2 d)0 - dA (14)dt m

From Sec. 2.18, 27A is the total magnetic flux linking a circle of radius r. Thus, with 2rAo definedas the flux linked by a surface on which the particles have no angular velocity, Eq. 14 can be integratedto obtain Busch's theorem:1

2 de (15)r - (A - A) (15)

This result is a useful integral of one of the equations of motion. It also lends immediate in-sight to the result of directing a beam of particles through a complex magnetic field, for if the beamenters from a field-free region with no angular velocity, so that Ao = 0, then it is clear that itleaves the magnetic field with no angular velocity.

11.3 Magnetron Electron Flow

Electron flow in a type of magnetron configuration illustrates the implications of the laws givenin Sec. 11.2. A uniform magnetic field, Bz, is imposed collinear with the axis of a cylindrical cathode,surrounded by a coaxial anode, as shown in Fig. 11.3.1. The arrangement is essentially that of a cyclo-tron-frequency magnetron, an early type of device for converting d-c energy (supplied by the source con-straining the anode to a potential V relative to the cathode) to microwave-frequency a-c.

Fig. 11.3.1.

In configuration of cyclotron-frequencymagnetron, electrons emitted from innercathode execute cyclotron motions asthey are accelerated toward anode acrossaxial magnetic field.

Secs. 11.2 & 11.3

1. J. R. Pierce, Theory and Design of Electron Beams, D. Van Nostrand Company, New York, 1949, p. 35.

11.3

Busch's theorem, Eq. 11.2.15, describes the tendency of the electrons to rotate about the magneticfield. Here, the flux density is uniform, so 2wA = wr2BZ . Also, the electrons have no angular velocityat r = a, so 2nAo = zb2Bz. Thus, Eq. 11.2.15 becomes

2de I bdt 2 c(1 ) (1)

r

where q = -e, and the electron cyclotron frequency is defined as wc = Bze/m.

An electron in the vicinity of the cathode is accelerated in the radial direction by the imposed

electric field. As its radial position increases, Eq. 1 shows that it picks up an angular velocity,

just as would be expected from the Lorentz force generated by the radial motion. The radial force equa-

tion is required to describe the trajectory. Because motions are in the steady state, the energy equa-tion, Eq. 11.2.10, provides a convenient first integral of this equation. The potential and kinetic

energies are both zero at the cathode, so that with the use of Eq. 1 for the angular velocity, it follows

that22rw 2

dr2 - 2 2e (2)+ 4 (I - 2 =

r

Thus, the electron executes radial motions in a potential well determined by the combination of the elec-

tric field tending to pull the electron outward and the magnetic field tending to divert it into anangular motion and eventually back toward the cathode. For the coaxial geometry, and in the absenceof space-charge effects, the potential distribution is

= V In ( )/ln (-) (3)

and so, Eq. 2 becomes an expression for the velocityas a function of radial position,

S 1 2(1 12 (4)dt 2 In a 2

r

where the normalization has been introduced,

r = r/b, a = a/b, t = twC

8 eV B e- 2m2 c m

w mbc

Typical potential wells are shown inFig. 11.3.2. For V = 10, the electron is returnedto the cathode while for V = 16 it collides withthe anode. For a critical value, V = Vc, the elec-tron just grazes the anode. This critical value isdetermined by setting Eq. 4 equal to zero withr = a,

V = (i - 2)a (6)c 2a

Integration of Eq. 4 gives

r2dr Fig. 11.3.2. Potential wells for cyclo-

fo In r 2 1 2 (7) tron motions. All variables are

VVIn a r ( normalized.r

Numerical integration gives the radial dependence shown in Fig. 11.3.3. In turn, the angular positionfollows from Eq. 1:

1 t (8)S (1 - )dt

o r

Sec. 11.3

I

11.4

0 I 2 3 4t-

Fig. 11.3.3. Radial position of electrons as functionof time. All variables are normalized.

The results from the radial integration can be used to numerically evaluate this expression and obtainthe trajectories shown in Fig. 11.3.4.

For these trajectories, where the potential is held fixed, the electron kinetic plus potentialenergy is conserved. With the introduction of a potential component varying at a frequency on the orderof wc, energy imparted to an electron by the d-c field can be removed in a-c form. With the d-c voltageadjusted to make V = Vc, the effect of a small increase in potential is dramatically different from thatof a small decrease. Suppose that as a given electron departs from the cathode, the potential increases.The electron is accelerated by the potential and hence takes energy from the source. But, it alsostrikes the anode or cathode and is removed after only one orbit. By contrast, an electron that leavesthe cathode as the potential is decreasing will be decelerated and hence give up energy to the a-csource. This electron does not strike the anode, and in fact tends to remain in the annulus for manycycles, contributing, along with electrons having a similar phase relation to the a-c field, to givingup energy to the a-c source.

If the a-c source is replaced by a low-loss resonator, the device can sustain self-oscillation.Hence, it can be used as a generator of energy having a frequency on the order of Wc. For electronswith Be = 0.1 tesla, wc/2w = 2.8 GHz. Common magnetrons make use of resonators to provide for atraveling-wave interaction with the gyrating electrons.1

1. H. J. Reich, P. F. Ordnung, H. L. Krauss and J. G. Skalnik, Microwave Theory and Techniques, D. VanNostrand Company, Princeton, N.J., pp. 708-735.

11.5 Sec. 11.3

Fig. 11.3..

Cyclotron motions in cylindricalmagnetron with normalized volt-age as a parameter.

11.4 Paraxial Ray Equation: Magnetic and Electric Lenses

Oscilloscopes and electron microscopes are devices exploiting electric and magnetic lenses.Simple lens configurations are shown in Fig. 11.4.1. In both the magnetic and electric configurations,the electron beam enters from a region where there is no magnetic field with an axial velocity which,according to the energy equation, Eq. 11.2.10, satisfies the relation

1 ,dz,2Im( r) = e (1)

The tendency of the axisymmetric magnetic field to focus the beam can be seen by considering theLorentz force on an electron entering the field somewhat off axis. The longitudinal velocity crosseswith the radial component of A to produce an angular velocity. Busch's theorem, Eq. 11.2.15, ex-ploits the solenoidal character of I to represent this effect of the rotational field in terms of theaxial field alone. Thus, even if the magnetic flux density near the axis is approximated as in-dependent of r, for an electron entering from a field-free region, Ao = 0, and the angular velocity canbe simply taken as

eBde = (2)dt 2m

This component in turn crosses with the axial component of B to deflect the electron toward the axis.Thus, while in the magnetic field, the electron is deflected toward the z axis; but, in accordance withEq. 2, once through the field it continues toward the axis without an angular velocity.

In the electric lens, the electron tends to be focused toward the axis as it enters the fringingfield, but to be diverted toward the electrode as it leaves the field. The net focusing effect derivesfrom the fringing field having a greater intensity as the electron enters than as it leaves.

Paraxial Ray Equation: An equation for the radial position, r, of an electron as a function of itslongitudinal position, z, is the basis for designing both magnetic and electric lenses. It pertainsto electrons traversing the fields near the axis where the magnetic flux density is essentially in-dependent of radius, i.e., of the form Bz(z). The radial component of i implied by the z dependenceis already built into Eq. 2. The radial component of t near the axis has an r dependence that can be

represented in terms of a given dependence of the potential 0 = O(z) at r - 0 by exploiting Gauss'law (Prob. 4.12.1):

2Er = e- (3)

r dz

Given Bz(z) and O(z), what is r(z)? With the use of Eqs. 2 and 3, the radial component of theforce equation, Eq. 11.2.3, becomes

2 eB 2dr 1 eBz2 er d2

2 -- ) r = m 2 2 (4)dt dz

Secs. 11.3 & 11.4 11.6

$

Fig. 11.4.1.

Fig. 11.4.1.

(a) Magnetic electron beam lens approximated by fieldsand focal length of Fig. 11.4.2.

(b) Electric electron beam lens withexemplified by Fig. 11.4.3.

trajectory

Here, the electron position is pictured with time as the independent parameter. With 4 and Bz independ-ent of time, the electron flow is steady so that time can be eliminated as a parameter and r = r(z).With the objective of writing Eq. 4 with z as the independent variable, observe that

dr dr dzdt dz dt

Sec. 11.411.7

and from the time derivative of Eq. I that

2d z - e d (6)dt2 m dz

In the lens region, Eq. 1 is approximate. With Eq. 6, it is therefore assumed that the longitudinalkinetic energy is much greater than that due to the radial and angular velocities. With the use ofEqs. 5 and 6 it follows that

2 2 2 2dr dr dz2 dr dz 2e d2r e dd drdt 2 d 2 ( ) + ( ) = - • + (7)2 2 dt dz 2 m 2 m dz dzdt dz dt dz

Thus, the radial component of the force equation, Eq. 4, becomes the paraxial ray equation,1

2dr + A dr + Cr = 0 (8)

2 dzdz

where

1 d 1 e 2 1 d2A = T- 7; B+2dz' = 8 m z 4 dz 2

Magnetic Lens: The limiting form of Eq. 8 for a purely magnetic lens is misleading in its sim-plicity (Prob. 11.4.1):

2dr 2

2 K r 0; K m B (9)dz

Through Eq. 1, 0 represents the incident axial velocity. A reasonable approximation to the on-axisaxial field for a solenoidal coil that is long compared to its radius is the distribution shown inFig. 11.4.2. Thus, in Eq. 9, Bz = 0 for z < 0 and z > k, and Bz = Bo over the length Z of the lens.An electron entering at the radius ro, with dr/dz - 0, therefore has the trajectory

r = r cos Kz (10)o

inside the lens and leaves on a straight-line trajectory with the slope

dr (z = ) = -r K sin K, (11)dz o

With the focal length, f, defined as shown in Fig. 11.4.2, it follows that

= (Ka sin KX) (12)

Thus, the focal length decreases with Bz (represented by Kt) as shown in Fig. 11.4.2.

Electric Lens: For numerical integration, Eq. 8 is written in terms of a pair of first-orderequations

d

wuh -Au - Cr

where z z/a, r = r/a, u = u, A = aA, and C = a2C.

As an example, consider the electric lens of Fig. 11.4.1. The potential distribution inside theabutting cylindrical electrodes with radius a that comprise the lens is found in Prob. 5.17.3 to be aninfinite series with radial dependences represented by Bessel functions. The z dependences of thedominant terms have an exponential decay away from the plane z = 0, exp Lz, where =i2.4 is the firstroot of the Besssel function Jo(a). For the present purposes, this longitudinal distribution of

1. J. R. Pierce, Theory and Design of Electron Beams, D. Van Nostrand Company, New York, 1949,pp. 72-91.

Sec. 11.4 11.8

I I

I-f--- zf

.5

0

Fig. 11.4.2. For magnetic lensBo over length k, focaldefined with Eq. 10.

0 I 5 0 15

of Fig. 11.4.1a, having essentially uniform axial fieldlength f normalized to k is given as a function of KZ,

- U I C U 'Ft U ( 0 Vz/a -

Fig. 11.4.3. For the electric lens of Fig. 11.4.1b, the axial potential distribution isrepresented by the broken curve. The solid curve is the electron trajectory pre-dicted by Eqs. 13 and 14 with Vo/V = 0.5 and B = 2.

potential is approximated by

- = V + - (1 + tanh ) (14)

This distribution of potential is shown in Fig. 11.4.3.

Using Eq. 14, A and C (given with Eq. 8) are given functions of z. Note that, although it has

space-varying coefficients, the paraxial ray equation, Eq. 8, is linear. Thus, general properties ofthe electron flow can be deduced using superposition. Numerical integration, using Eqs. 13, is straight-forward and results in electron trajectories typified in Fig. 11.4.3. Note that the electron velocity,

deduced at any given point from Eq. 1, is increased in the conservative transition through the lens.

Sec. 11.4

/I

11.9

11.5 Plasma Electrons and Electron Beams

A model often used to represent electronic motions in a "cold" plasma, and even electron beams in"vacuum," gives the electrons a background of ions that neutralize the space charge. Because the ionsare much more massive than the electrons, on time scales of interest for the electron motions, the ionsremain essentially motionless.

A uniform axial magnetic field is imposed. In equilibrium, the electrons stream with a uniformvelocity U along the magnetic field lines. Electron motions across the magnetic field result in cyclo-tron orbits that tend to confine the motions to the axial direction.

Because the electron motions are axial, the transverse components of the force equation only givean after-the-fact approximation to the transverse components of the velocity. The axial component ofthe force equation is, to linear terms in the velocity - = (vz + U)Tz,(avav

m + U(1)at az m az

The current density for electrons having a number density no+n(x,y,z,t) and this velocity is

+ + 4-J = -enoUiz - e(nU + nvz)iz (2)

so that to linear terms, conservation of charge requires that

-- (enU + envz) - (ne 0 (3)z0 z at

Finally, because the equilibrium electronic space-charge density, -noe, is cancelled by that due topositive ions, Gauss' law requires that

V2( = ne (4)E

Perturbations vz , n and 4 are described by Eqs. 1, 3 and 4.

Transfer Relations: Consider now a planar layer of plasma or beam having thickness A in thex direction. Solutions to Eqs. 1 and 3 take the form vz = Revz(x) exp j(t - kyy - kzz ) , so substitu-tion into Eq. 1 gives

e= (5)z=- (5)

In turn, Eq. 3 and this result give

^ k2knv kneZOZ _ Z An 0 z 0 (6)

(w - kU) (_ 2

Finally, this relation combines with Eq. 4 to show that the potential distribution must satisfy

2- =0; y = k + k p (7)y z(dx--d-

-2'where the plasma frequency is defined as w = In e /e m. This relation is of the same form as for re-presenting Laplace's equation in Sec. 2.16 Thus, the transfer relations are the same as inTable 2.16.1, provided that y is defined as in Eq. 7. Note that a similar derivation leads to trans-fer relations in cylindrical geometry so that the ttansfer relations for an annular beam are as givenby the relations of Table 2.16.2 with coefficients suitably defined. For example,fm(x,y) -+fm(x,y,k - y), where y may differ from one annular region to another.

Space-Charge Dynamics: The sheet beam shown in Fig. 11.5.1 exemplifies the dynamics of beams inuniform structures. The surrounding region is free space, with walls to either side constrained by atraveling wave of potential. The wall potential can be regarded as a given drive, although moregenerally it can be made consistent with external electromagnetic structures. The velocity is purelyaxial, so the boundaries do not deform and boundary conditions are simply

Sec. 11.5 11.10

Fig. 11.5.1

U Planar electron beam in uniformaxial magnetic field.

-- ------------------------4

c = Vo, 4d= ;e, D = De, = 0 (8)x x x

Here, interest is restricted to motions that are even in the potential, so with the last boundary con-dition it is presumed that @C = $i.

Transfer relations for the free-space region and for the beam follow from Eq. 7 and Table 2.16.1:

cIjc 1DC -coth ka sinh ka

x sinh ka= ek (9)

dI Icoth kax sinh ka

sx snh yb

1f0inh yb coth yb

The last equation gives ;f in terms of ;d. This is inserted into Eqs. 9b and 10a, set equal to eachother. The resulting expression can be solved for Od. Substituted into Eq. 9a, that gives

S= -k (k + Ycth ka tanh b) c; D k coth ka + y tanh yb (11)x D(w,k)

This result gives the driven response, but also embodies the temporal modes and spatial modes, asdiscussed in Secs. 5.15 and 5.17. These are described by the dispersion equation, D(w,k) = 0.

Temporal Modes: It is clear that there are no roots of this expression having 7 purely real.Purely imaginary roots abound, as is evident by substituting y - ja:

(ab)tan(ab) = (kb)coth[(kb) a] (12)

Graphical solution of this expression, for a given a/b, results in roots, an . The frequencies of associ-ated temporal modes follow from the definition of y2= -a2 given with Eq. 7:

S U(b 1/2W = bk (-)+ (13)Zp L) (bk) -1/2

Here it has been assumed that ky = 0. This dispersion equation is represented graphically byFig. 11.5.2.

For each real wavenumber, k, there are two eigenfrequencies representing space-charge waves. Inthe absence of convection these have phase velocities, m/k, in the positive and negative directions.With convection, these waves are respectively the fast and slow space-charge waves that are central toa variety of electron-beam devices and interactions. The transverse dependence of a temporal modehaving a real wavenumber kz = k is sinusoidal within the beam and exponential in the surrounding regionsof free space, as depicted by the inserts to Fig. 11.5.2.

Without the equilibrium streaming, the temporal modes are similar to those of the internal electro-hydrodynamic space-charge waves of Sec. 8.18. There are an infinite number of modes, n > p, withinthe region of the u-k plot bounded by the fast and slow wave branches for any given mode n = p. In the

Sec. 11.511.11

3

2WP

I

0 I 2 3kb-

Fig. 11.5.2. Normalized angular frequency as a function of normalized wave-number for space-charge waves on planar electron beam of Fig. 11.5.1.Inserts show transverse distribution of potential for two lowesteigenmodes at longitudinal wavenumber kb = 1.

frame of reference moving with the beam velocity U, the frequencies, w-kU, of these modes approachzero as the mode number, and hence the number of oscillations over the transverse dimension of thebeam, approaches infinity.

Spatial Modes: Typically, in electron beam devices, it is the response to a given driving fre-quency that is of interest. The homogeneous part of the response is made up of spatial modes havingwavenumbers that are solutions to D(w,k) = 0, with w the specified driving frequency. In general, theseare complex roots of a complex equation. However, those modes having real wavenumbers for the realdriving frequency can be identified from Fig. 11.5.2.

Sec. 11.5

'4

11.12

DYNAMICS IN SPACE AND TIME

11.6 Method of Characteristics

In representing the evolution of a continuum of particles, it is natural to express the partialdifferential equations of motion as ordinary equations with time as the independent variable. Asillustrated in Secs. 5.3, 5.6 and 5.10, what can result is a complete picture of the temporal evolu-tion, but one viewed along a characteristic line in (T,t) space. The price paid for the character-istic formulation is an implicit dependence on space. That the characteristic lines do not have to beidentified with particles is illustrated in this and the next five sections. Physically, the character-istics now represent waves rather than particles. However, the objective is again to reduce partialdifferential equations to ordinary ones.

If the equations, written as a system of first-order expressions, have coefficients that are notfunctions of the independent variables, they are said to be quasi-linear. An example comes from theone-dimensional longitudinal motions of a highly compressible gas.

For now, there is no external force density and viscous effects are ignored. Thus, with theassumption that 9 = v(z,t)tz, p = p(z,t) and p = p(z,t), the force equation, Eq. 7.16.6, becomes

av av 0 (1)p(-t + v •7 ) + = 0 (1)

The flow is not only assumed adiabatic, but initiated in such a way that every fluid particle can betraced backward in time along a particle line to a point in space and time when it had the same state

(p,p) = (po,Po). The flow is initiated from a uniform state. Thus, Eq. 7.23.13 holds throughout the

region of interest, and it follows that

S=a2 aP; a op (2)=az a ) 1 (2)

o o

It follows that Eq. 1 can be written as

(0) -v + (a 2 ) •p + (P) a- + ) a = 0 (3)at as at as

Conservation of mass, Eq. 7.2.3, provides the second equation in (v,p):

(1) + (v) + (0) + (p) 0 (4)at az at az

These last two expressions typify systems of first-order partial differential equations with two in-dependent variables (z,t). They are not linear, but do have coefficients depending only on thedependent variables (v,p). The characteristic equations are now deduced following the reasoning ofCourant and Friedrichs.

1

First Characteristic Equations: Arbitrary incremental changes in the time and position result inchanges in (p,v) given by

dp = -p dt + ap dz (5)at az

Bv avdv = -v dt + v dz (6)at az

The objective now is to find a linear combination of Eqs. 3 and 4 that takes the form

f(P)dp + g(v)dv = 0 (7)

because this equation can be integrated. To this end, note that a line in the z-t plane along whichdp and dv are to be evaluated has not yet been specified. It can be selected to guarantee the desiredform of the equations of motion.

1. R. Courant and K. 0. Friedrichs, Supersonic Flow and Shock Waves, Interscience Publishers,New York, 1948, pp. 40-45.

Sec. 11.611.13

A linear combination of Eqs. 3 and 4 is written by multiplying Eq. 3 by the parameter XA.andEq. 4 by XAand taking the sum:

Ba p av+ avLy + 2 y v-t(1)2 (Y+ a ) +lP + 2vP+) = 0 (8)

If this expression is to have the same form as Eq. 7, where dp and dv are given by Eqs. 5 and 6, then

2dz_ 1V + A2a dz X1 P

+ X2vpdt ; t p(9)

These expressions are linear and homogeneous in the coefficients (p,v):

dz 2 1--v -a I 0dz = (10)

jj dt 2 0

It follows that if the coefficients are to be finite, the determinant of the coefficients must vanish.Thus,

+dzd- = v + a; C (11)dt -

If v,p and hence a(p) were known functions of (z,t), these expressions could be solved to give familiesof curves along which Eq. 8 would take the form of Eq. 7. Apparently two such families have been found.They are called the Ist characteristic equations and respectively designated by C+ and C-.

Differential equations, such as Eqs. 3 and 4, for which the Ist characteristic equations arereal, are said to be hyperbolic. Elliptic equations, for which the Ist characteristics are not real,must be solved by some other method than now described.

Second Characteristic Lines: The goal of writing Eq. 8 in the form of Eq. 7 is achieved byfactoring X1 from the first two terms and A2 from the third and fourth terms, and then substitutingfor the coefficients of the second and fourth terms, respectively, using Eqs. 5 and 6:

Xp 9p dz av 3v dz1A t + z dtas dt+) = 0 (12)

If this expression is multiplied by dt and divided by A1 , the desired form follows:

2dp + p -- dv = 0 (13)

To establish the ratio 12 /l1 , either of Eqs. 10 can be used. For example, substituting 12 /X1 as foundfrom Eq. 10a gives

dp + (! - v)dv = 0 (14)2 -dta

This expression is further simplified by using the Ist characteristic equations, Eqs. 11, to write

+a

dv + dp = 0 on C (15)-P

where the choice of signs is determined by which sign is being used in Eqs. 11.

With a(p) specified by Eq. 2, the IInd characteristic equations can be integrated:

++ 2a(p=c+ on C (16)

Here, c+ and c- are respectively invariants along the C+ and C- characteristic lines.

Systems of First-Order Equations: The method used to determine the first and second character-istic equations makes their deduction a logical response to the objective. As long as the number ofindependent variables (z,t) remains only two, the same technique can be used with more complex problems.

Sec. 11.6 11.14

But, it is convenient in dealing with several first order equations to use a more formal approach tofinding the.characteristics. Although the formalism now considered appears to be different, in factthe characteristic equations are the same.

Equations 3 and 4 are particular cases of the first two expressions in the set of four,

C

D

dp

dv

A1 A2 A3 A4

BI B2 B3 B4

dt dz 0 0

0 0 dt dz

ap

apazByvBtavaz

(17)

Here, the coefficients Ai and Bi are in general functions of (p,v,z,t). Also, for generality,C = C(p,v,z,t) and D = Dtp,v,z,t) represent the possibility that the differential equations are in-homogeneous in the sense that they have terms which do not involve partial derivatives. The last twoexpressions will be recognized as the differential relations for dp and dv, Eqs. 5 and 6.

Following the formalism leading to Eqs. 11 and 15, Eqs. 17a and 17b are multiplied respectivelyby 1I and 12, and added. Then the ratio of coefficients for the respective Z/at's and 2/az's arerequired to be dz/dt, and the result is two homogeneous equations in the V's:

= 0 (18)

The first characteristic equations are found by requiring that the determinant of the coefficients inEq. 18 vanish.

This same condition is obtained by requiring that the determinant of the coefficients in Eq. 17vanish. To see this, rows three and four are multiplied by (dt)-1 . Then rows three and four aremultiplied respectively by -A1 and -A3 and added to row one. Similarly, rows three and four can bemultiplied respectively by -B1 and -B3 and added to row two. The result is the determinant

dz dz0 A -A, 0 A-A--

A2 - .dt 4 3 ddz dz

0 B -B Lz 0 B -B -2 1 dt 4 3 dt

dt dz 0 0

0 0 dt dz

= 0 (19)

Now, by expanding about the dt's that appear in columns with all other entries zero, the same require-ment as given by Eq,. 18 is obtained. The first characteristic equations are obtained by writing thedifferential equations in the form of Eq. 17 and simply requiring that the determinant of the coeffi-cients vanish. The same approach can be used with an arbitrary number of dependent variables.

To solve Eqs. 17 for any one of the four partial differentials would require substituting thecolumn on the left for the column of the square matrix corresponding to the desired partial derivative,and to divide the determinant of the resulting matrix by the coefficient determinant. However, thecoefficient determinant has already been required to vanish, since that is just the condition for ob-taining the Ist characteristic equations. Thus, if the partial derivatives are not infinite, thenumerator determinants must also vanish. Four equations result which are reducible to the IInd charac-teristic equations.

As an example, consider once again Eqs. 3 and 4. The coefficient determinant is

1 v 0 p

0 a2 P pv 0 (20)

dt dz 0 0

O 0 dt dz

111

Sc 1.

11.15 Sec. 11.6

and can be expanded, taking advantage of the zeros, to give Eqs. 11. Then the numerator determinantfor finding ap/at is the coefficient matrix with the column matrix on the left in Eq. 17 substitutedfor the first column on the right, or

r0 v 0 p

20 a p pv

dp dz 0 0

dv 0 dt dz

= 0 (21)

With the use of the first characteristic equations, Eq. 21 reduces to the second characteristic equa-tions given by Eqs. 15. A check of the other three equations obtained by substituting the columnmatrix in the second, third and fourth columns gives the same result.

11.7 Nonlinear Acoustic Dynamics: Shock Formation

The longitudinal motions of a gas under adiabatic conditions both serve as a vehicle for seeinghow the characteristic equations are used, and provide insight into the nonlinear phenomena that areresponsible for wave steepening and shock formation.1 (See Reference 10, Appendix C.)

Initial Value Problem: The characteristic equations are given by Eqs. 11.6.11 and 11.6.16.Although there is no necessity for linearizing, it is helpful to realize how perturbations from a uni-form flow with velocity U, density po and acoustic velocity ao are represented by the characteristics.(The linearized versions of Eqs. 11.6.3 and 11.6.4 are the one-dimensional forms of Eqs. 7.11.1-7.11.3.)In that limit, the first characteristic equations have U + ao on the right, and are therefore inte-grable to give straight lines with slopes equal to the wave velocities U + ao in the +z directions.These families of lines are illustrated in Fig. 11.7.1a.

In general, v and a can vary and the characteristic lines sketched in Fig. 11.7.1b in the z-tplane are not known. However, the functions v and a(p) are known at an intersection of the C+ and C-characteristics, wherever that may be. That is, suppose the initial values of v and a at points A andB shown in Fig. 11.7.2 are given (at t = 0 but at different points along the z axis). Then, fromEq. 11.6.16a, c+ is

a r2a(p)+ = [v]A + [ _I (1)

Similarly, from the initial conditions at B,

b2a() (2)cb = [V]B - [y( (2)B

a + bNow, c+ is invariant along the C characteristic, and c- is invariant along the C characteristic. Atpoint C, certainly at a later time and generally at a different point in space than either A or B,Eqs. 11.6.16 both hold, with c+ = c and c- = c,. Hence, they can be solved simultaneously for eitherv or a(p). For the former, addition yields

a bc+ c

[V]C 2 (3)

Given conditions at A and B, the solution at C is established. The solution.is known, but whereand when does it apply? The Ist characteristic equations must be integrated to determine the locationof C in the z-t plane.

The characteristic lines have the physical interpretation of being wave fronts. Conditions at Aand B propagate along the respective lines and combine at C to give the response. It is remarkablethat what happens at C depends only on the conditions at A and B. But the "location" of C is determinedby initial conditions everywhere between A and B, as can be seen by considering how a computer can beused to "march" from left to right in the z-t plane and determine the solution in a stepwise fashion.

Secs. 11.6 & 11.7

1. R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves, Interscience Publishers,New York, 1948, pp. 40-45.

11.16

C+

C C4

(a) t (b) t

Fig. 11.7.1. (a)Linear case where v and a are known constants and the charac-teristics are straight lines. (b) c4 and cý are established from theinitial conditions at A and B. Because they are invariant along the C+

and C- characteristics respectively, the solution is established wherethe characteristics intersect at C.

The Response to Initial Conditions: In the linearized case, the characteristics are straightlines as shown in Fig. 11.7.1a. The point of intersection, C, is then determined because the z co-ordinates of A and B, as well as the characteristic slopes, are known. The effect of the non-linearity is to bend the characteristic lines in the z-t plane. This is not surprising, because itwould be expected that the velocity of propagation of wavefronts (the slope of dz/dt of a character-istic line) depends on the local speed of sound superimposed on the local velocity of the fluid.

7In Fig. 11.7.2, a discrete representation of the charac- L

teristics is made. Initial values are given at the positionsz = zi. The C+ line emanating from z and the C- line from zkcross at some point (j,k). To find the solution throughout thez-t plane, k-I

a) Evaluate the invariants using the initial values in

Eqs. 1 and 2:

C+= [v] ± [(4)

b) Tabulate solutions at all intersections (j,k) bysolving simultaneously Eqs. 4:

t1[v] (j)= (c + ck )

+(5) Fig. 11.7.2. The (z-t) intersection

of jth C+ characteristic and

[a] (j) = (c- c k ) -1 kth C- characteristic is de-(j,k) + 2 noted by (j,k).

c)Use the results of (b) to tabulate all characteristic slopes at the intersections (j,k):

d +[z1- = [v] + [a] (6)dt ( ,k) (j,k) (,k)

d) Start when t = 0 and build up grid by approximating characteristic lines as being straightbetween points of intersection. Coordinates and slopes at neighboring points (zj k-1) and (zj+lk)determine z-t coordinates of point (zj,k). Thus both the solution and the z-t coordinate at which itapplies are determined.

Sec. 11.7

C•I%ko

11.17

Simple Waves: Initial and boundary conditions are illustrated in Fig. 11.7.3 in which the fluidis initially static [v = 0, a(p) = a] and is driven by a piston at one end. The piston, with positionshown as a function of time in the figure, is initially at rest at z = 0 and is pushed into the gasuntil it reaches the final position z = zo . The slope of the piston trajectory has the physical sig-nificance of being the piston velocity; hence the velocity of the fluid along the piston trajectory isthe slope of the trajectory, (dz/dt)p.

( ((Kh\

I,4-,

\uJ /

Fig. 11.7.3. (a) Gas filled tube driven by piston. (b) Boundary and initial valueproblem where the initial state of the fluid is uniform (when t = 0) and an ex-citation is applied by means of a piston which is initially at z = 0. (c) a andv are constant along C+ characteristics, which are straight lines.

By definition, the initial boundary value problem described leads to simple-wave motions. Thisname designates the response to a boundary condition with the region of interest having a uniforminitial state.

Consider the two C characteristics sketched in Fig. 11.7.3b. They intersect the z-axis atpoints A and B, where the initial conditions require that the fluid is stationary (v = 0), and thatthe velocity of sound is ao . From this, it follows that the invariants c., established at point Aand at point B using Eq. 11.6.16, are the same,

-2aa b o

c = c (7)- - y-1

Points C and D are intersections with the same C+ characteristic. Hence, the invariant c+ is thesame at points C and D. Given c+ and c_ along the characteristics which intersect at C, the velocityand density at that point are found by simultaneously solving Eqs. 11.6.16,

c+ + cav(C) = 2 (8)

Similarly,

bc+ + cb

v(D) = 2 (9)2

a bHowever, because of the special nature of the initial conditions, Eq. 7 requires that c = ca , and itfollows that v, and by similar arguments, a(p) or p, are constant along any given C+ characteristic.Even more, because v and a are constant, it then follows from Eqs. 11.6.11 that the C+ characteristicshave constant slope.

The C+ characteristics appear as shown in Fig. 11.7.3c. Along the characteristics shown, thevelocity, v, remains equal to that of the fluid at the piston (point B), where

dzv =(q-)p (10)

r. ,

Sec. 11. 7 11.18

Z

0 4 8 12 16t--- l

Fig. 11.7.4. Simple-wave characteristic lines initiated by piston.

4 8 12 16 20 t

Fig. 11.7.5. Velocity of fluid as a function of z and t.

With Eq. 7, c- is established along the C- characteristic, and it follows from Eq. 11.6.16 that thesound velocity a(p) at the point B on the piston surface, where v is (dz/dt)p, is

a = a + -) (11)2 (•• p

This velocity of sound, a, along with the implied density p and velocity v from Eq. 10, remain constantalong the C+ characteristic. The picture of the dynamics is now complete, because the C+ character-istic emerging from B has a constant slope given by the Ist characteristic equation, Eq. 11.6 .11:

dz dz (+)d(it- + (-) p 2dt 0 dao

(12)

Suppose that the piston position depends on time, as shown in Figs. 11.7.4 and 11.7.5. With noinstantaneous change in velocity when t = 0, the piston reaches a maximum velocity when t = 3, and thendecelerates to zero velocity by the time t = 6. The C+ characteristics originating on the piston canbe plotted directly using Eq. 12. Here, it is assumed for convenience in making the drawing that y and

Sec. 11.711.19

ao are unity. Note that as the piston velocity increases, the characteristic lines increase in slope,while characteristics originating from the piston when it decelerates decrease in slope. Rememberthat the fluid velocity v at the surface of the piston is just the slope of the piston trajectory.This velocit'y remains constant along any given Cr characteristic, Hence, a plot of the fluid velocityas a function of (z,t) appears as shown in Fig. 11.7.5. In regions where the characteristics tend tocross, the waveform tends to steepen, until at points in the z-t plane where the characteristics cross,the velocity becomes discontinuous. This discontinuity in the wavefront is referred to as a shockwave. With the steepening, variables change more and more rapidly in space. This shortening ofcharacteristic lengths brings into play phenomena not included in the adiabatic model.

Note that a shock wave tends to form from a compression of the gas. By contrast, the decelara-tion of the piston tends to produce a waveform which smoothes out. Fluid near the leading edge of thepulse is moving in the positive z direction, and this adds to the velocity of a perturbation, a, in thatregion. Hence, variables within the pulse tend to propagate more rapidly than those nearer the leadingedge, and the wave steepens at the leading edge. Similar arguments can be used to explain thesmoothing out of the pulse at the trailing edge. In any actual situation, y will exceed unity, and theincrease in density, and hence acoustic velocity, makes a further contribution toward the nonlineareffect of shock formation. In actuality, effects of viscosity and heat conduction prevent the forma-tion of a perfectly abrupt discontinuity in p, a, and v.

Limitation of the Linearized Model: To be quantitative in giving conditions under which non-linearities are important, suppose that the piston is set into motion when t = 0 and reaches thevelocity (dz/dt)p by the time t = T. Then the characteristics are essentially as shown in Fig. 11.7.6,where the displacement of the piston is ignored compared to other lengths of interest. From Eq. 12,the characteristic originating at t = 0, z = 0, is

z = a t (13)

while that originating at t = T, z = 0 is

z = [a0 + ()p (')](t - T) (14)

Nonlinear effects will be important at z = £,where these characteristics cross. Solving Eqs. 13 and14 simultaneously for z = .by eliminating t gives

F ad= aT +1 (15)

Hence, for a given characteristic time T, say the period in a sinusoidally excited system, there is alength (some fraction of £) over which a linear model gives an adequate prediction. This distancebecomes large as (dz/dt)p becomes small compared to the velocity of sound, ao . It is clear, also, thatmaking the period small (the frequency high) can also lead to nonlinearities. This fact is not aslimiting as it seems, since the peak piston velocity in any real system is likely also to decrease asthe frequency is increased.

Fig. 11.7.6

An excitation at z = 0 raises thefluid velocity from 0 to (dz/dt)pin the characteristic time, T.Then nonlinear effects become im-portant in the distance t requiredfor the resulting characteristicsto cross.

Sec. 11.7 11.20

11.8 Nonlinear Magneto-Acoustic Dynamics

The longitudinal motions of a perfectly conducting gas stressed by a transverse magnetic field,discussed in Sec. 8.8 for small perturbations of a slightly compressible fluid, provide an example ofnonlinear electromechanical waves. The methods of Secs. 11.6. and 11.7 are put to work in many in-vestigations of magnetohydrodynamic waves and shocks, especially in the limit of perfect conductivityconsidered here.1 Motions, considered here perpendicular to the imposed magnetic field, have been con-sidered for arbitrary orientations of the field.2

Equations of Motion: At the outset it is assumed that the motions are one-dimensional:

v = v(z,t)T ; H = H(z,t)i (1)

The physical laws governing the dynamics are those of compressible fluid flow and magnetoquasi-statics. Reduced to one-dimensional form, conservation of mass, Eq. 7.2.3, requires that

~a+v -- + p -~= 0 (2)ýt 5z az

In writing the force equation, Eq. 7.16.6, viscous forces are ignored. The magnetic force density isconveniently written by using the stress tensor, Eq. 3.8.14 of Table 3.10.1;

av av p = aHp ( + v _) +v = -P H -z (3)

In the perfectly conducting fluid, there is by definition no electrical dissipation. If in additioneffects of dissipation and heat conduction are negligible, the energy equation, Eq. 7.23.3, reducesto an expression representing an isentropic process, Eq. 7.23.7:

tt(pp - ) = ( + v z-)(pp- ) (4)

In view of the one-dimensional approximation and Eq. 1, the field automatically has zero divergence.The combination of the laws of Ohm, Faraday and Ampere are represented by Eq. 6.2.3. The x componentof that equation, in the limit where a + m, becomes

-(vH) +_ = 0 (5)

The other components of Eqs. 3 and 5 are automatically satisfied.

Characteristic Equations: Following the technique outlined in Sec. 11.7, Eqs. 2-5, together withthe relations between changes in the dependent variables along the characteristic lines and the partialderivatives, are arranged in a matrix. For convenience, Dp/3t - P,t, ap/Bz = P'z, etc.:

0

0

0

0

dp

dv

dp

dHJ

1 v 0 p 0 0 0 0

0 0 p pv 0 1 0 p0H

-yp -ypv 0 0 p pv 0 0

0 0 0 H 0 0 1 v

dt dz 0 0 0 0 0 0

0 0 dt dz 0 0 0 0

0 0 0 0 dt dz 0 0

0 0 0 0 0 0 dt dz

p,t

P'z

v, t

V,z

p,t

p,'

H,t

H,z

(6)

1. G. W. Sutton and A. Sherman, Engineering Magnetohydrodynamics, McGraw-Hill Book Company, New York,1965, pp. 309-339.

2. W. F. Hughes and F. J. Young, The Electromagnetodynamics of Fluids, John Wiley & Sons, New York,1966, pp. 312-318.

Sec. 11.811.21

To obtain the Ist characteristic equations, the determinant of the coefficients is required tovanish. The determinant is reduced by following steps similar to those that lead from Eq. 11.6.17 to11.6.19, and then expanding by minors:

dz =v on Cdtdt 112 1/2 (7)

dt v + % on C-; % -[ +

The Cp characteristics are the particle lines, and actually represent two degenerate sets of character-istics. The CG characteristics represent magneto-acoustic waves, discussed for small amplitudes inSec. 8.8.

The second characteristic equations are obtained from Eq. 6 by solving the determinant arrived atby substituting the column on the left into the first column of the square matrix. Straightforward ex-pansion gives

(dz \{dE4dz, dzI+d dz 2 +Ypv+( -v)dv P( - v) +dp - - v)

dz dz-dp - [ H -]dH} = 0 (8)

The second characteristic equations along the particle lines are found from Eq. 8 using Eq. 7a. Thatthe determinantal equations are degenerate is again reflected by the first factor in Eq. 8; rememberthat (dz/dt - v) appeared as a quadratic factor in the denominator. The second term in brackets iszero if dz/dt = v, so that

[ dp - dp] + [dp dH] = 0 on Cp (9)p o0 p H

This equation is actually the sum of two independent expressions, as can be seen by considering Eq. 4,which on the particle characteristic can be written as

1- dp - dp = 0 or d(pp-Y ) = 0 on Cp (10)P

On the same particle characteristics, Eqs. 2 and 5 combine to give

- H= 0 or d() = 0 on Cp (11)p H p

These last two equations insure that 9 is satisfied, and account for the degeneracy of the two character-

istic equations.

Using Eqs. 7b in 8 gives the two additional characteristic equations

-- +

+pabdv - dp - poHdH = 0 on C± (12)

Thus, the Ist characteristic equations are summarized by 7 and the IInd characteristic equations givenby Eqs. 10-12.

Initial Value Response: To any given point in the (z,t) plane can be ascribed four intersectingcharacteristic lines, two of which are simply the particle line. These are illustrated in Fig. 11.8.1.In general, the solution at the given point is obtained by simultaneously solving Eqs. 10-12, which arefour equations in four unknowns. The first two of these expressions can simply be integrated to giveinvariants along cP:

pp-Y ppY on C (13)

c cH/p = Hc /P c on Cp (14)

The second of these states that a fluid circuit of fixed identity must conserve magnetic flux.Hence, an increase in density caused by the compression of a fluid element is accompanied by a localincrease in H. The model of Fig. 8.8.1a remains a useful way of viewing the interaction.

Sec. 11.8 11.22

Suppose that, at some time in the evolution of the system,the pressure, density and field are uniform and are (po,po,Ho).The invariants on the right in Eqs. 13 and 14 are independentof position thereafter:

-Y -Ypp = poo - Y

H Ho

P Po

(15)

(16)

It follows that the remaining IInd characteristic equations,Eqs. 12, become

ab H F- dp + dv = 0; ab = [oPoY(PY l) + po()2] (17)

op

In effect, the dynamics now involve only the characteristics C-and two of the four original variables. Given (p,v) from solvingEqs. 7b and 17, p and H are found from 15 and 16. The dynamics aresimilar to those for the gas alone, except that a + ab(p). Note

t)ig. 11.8.1. The solution at (z,t)

results from invariantscarried along the four char-acteristic lines shown, withCP representing two familiesof characteristics.

that the dependence of the magneto-acoustic velocity on p is in part determined by Ho .

11.9 Nonlinear Electron Beam Dynamics

The nonlinear motions of streaming electrons are usually described in terms of Lagrangian co-ordinates. Nevertheless, an Eulerian description affords considerable insight if it is couched interms of characteristics. By contrast with the equations of Secs. 11.7 and 11.8, those now consideredare inhomogeneous.

The laws describing an electron beam neutralized by a background of ions are of the same nature asused in Sec. 11.5. Here, they are written without linearization but with the assumption that motionsand fields are one-dimensional and that fields, like the motion, are z-directed. Hence, particle con-servation requires that

Sz an an(no + n ) + a + vz - z= 0

where no is the equilibrium number density and n(z,t) is the departure from that equilibrium. The longi-tudinal force equation is

av

at

avz _ e

z a. m z

and in one dimension, Gauss' law is

aEz en

az E

Variables are normalized at the outset so that timeWp4-e 2 no/mco,

t = t; z = z/R; n = n/no;

E= E /(n ei/ o)

By definition,

o ave z=-

and Eq. 3 becomes

DEn = --az

is measured in terms of the plasma frequency

o oe=e/W ; v v/(zw)p

Secs. 11.8 & 11.911.23

Then, Eq. 1 and D( )/az of Eq. 2, combined with Eq. 3, are the first two of the four equations,

1 v 0 0

0 0 1 v

dt dz 0 0

0 0 dt dz

anatanBzBt

Bzaz

-(1 + n)8

n -

dn

ode

(7)

The third and fourth are expressions for dn and de, introduced following the procedure described inSec. 11.6.

That the determinant of the coefficients vanish gives the Ist characteristic equations. Thereare two families of lines, but these are degenerate:

dz +d- = v on C- (8)dt

The second characteristic equations could be obtained by substituting the column on the right for anypair of columns on the left and setting the respective determinants equal to zero:

dn 0 +dt = -(1 + on C (9)dt 2

de n - 2 on C (10)

These expressions are simple enough that they could have been obtained directly from Eqs. 7a and 7b byinspection.

A configuration typical of klystron beam-cavity interactions is shown in Fig. 11.9.1. The elec-tron beam passes through screen electrodes at z = 0 and z = £. These are constrained to the potentialdifference, V(t) = V (t)(noeZ2 /co), which will be taken here as a given drive. In reality, V(t) mightbe associated with a resonator that is used to either excite the beam or extract energy.

The region of interest in the z-t plane is between the electrodes, 0 < z < A and for 0 < t.Characteristic lines that enter this region along the z-axis (when t = 0) are denoted by K = N...M,while those that enter along the t-axis (where z = 0) are represented by K = 1...N. To integrateEqs. 9 and 10, it is appropriate to have two initial conditions for the latter and two entrance bound-ary conditions for the former.

When t = 0, the velocity and electric field distributions between the screens are taken as known.As an example, if the beam is initially unmodulated, the electron velocity is constant and there is nospace charge between the screens:

v = U, E = V(O) (11)

For the boundary conditions, it is assumed that the beam enters with a constant velocity, U. Inpassing through the screen, an electron is subjected to a step in electric field, but not to an impulse.Hence, this velocity is continuous through the screen, this means that along the t-axis, where theelectrons enter the region of interest, av/at is also continuous. It follows from the force equationfor an electron; Eq. 2, that 9 is not continuous through the screen. Rather, if 0= 0 just upstream ofthe screen at z = 0, according to Eq. 2, e assumes the value

(0,t) = - E(,t) (12)U (2)

just downstream. The number density is continuous through the screen, so that if the beam is un-modulated upstream, then just downstream

n(O,t) = 0 (13)

Although not relevant as a boundary condition, it can be seen from Eq. 1 that the step in e across thescreen is accompanied by a step in an/az.

To make Eq. 12 a useful boundary condition, E(0,t) must be related to V(t). To this end, twointegrations of Eq. 6, with the condition that the second intergration give V(t), result in

Sec. 11.9 11.24

i

+ I I

d;

uli?? i ?T L= 2 "'twp

(a)Fig. 11.9.1. (a) Beam enters region between screen electrodes

frequencies, the screens typically provide coupling tothe inset. (b) Characteristic lines in the z-t plane.characteristic line (K) and time (L).

L L=N

(b)with velocity U. At microwavecavity resonators, as shown byCoordinate (x,t) is denoted by

E = V - JJ n(z',t)dz'dz + J n(z',t)dz'0o o

so that the electric field at z = 0 can be evaluated and used to express Eq. 12 as

(0,t) = - + n(z',t)d'dzU U jo

Equations 13 and 15 comprise the boundary conditions at z = 0.

(14)

(15)

The integration of the second characteristic equations, Eqs. 9 and 10, can be carried out bytreating them as simultaneous ordinary differential equations. This is possible only because thecharacteristic lines to which they apply are the same. Depending on whether the characteristic line ofinterest enters through the t = 0 axis or the z = 0 axis, the initial conditions or boundary conditionsserve as "initial" conditions for this integration. However, the integration is not quite this straight-forward, because superimposed on the propagational dynamics is Poisson's equation, which makes theentrance field instdntaneously reflect both the net effect of the charge in the region of interest andthe voltage V(t). This is why the boundary condition on 8, Eq. 15, depends not only on the voltage butalso on the charge throughout.

Consider the numerical steps that portray the space-time dynamics while marching forward in time.The initial conditions when t = 0 (L = 1) can be used with Eqs. 9 and 10 to establish n(z,dt) = n(K,2)and 8(z,dt) E O(K,2) at the points where the characteristics (K = N.. M) intersect the t = dt axis (L=2).Also, Eqs. 8 can be used to determine where these solutions apply, i.e., where

z

v(z,dt) = U + e(z,O)dzo

(16)

Sec. 11.9

---- - -

( t) I\

11.25

Fig. 11.9.2. Turn-on transient in configuration of Fig. 11.9.1 with sinusoidal voltage applied toscreens. Normalized U = 2, V = 2 and angular frequency w = (2pw ).

The characteristic line K = N-1 entering at z = 0 when t = dt does so with conditions set by theboundary conditions of Eqs. 13 and 15. Note that because n(0,dt) E n(N-1,2) is known and n(z,dt) hasalready been determined at the location K = N--.,L = 2, integration called for in Eq. 15 can be carriedout. Hence, (0O,dt) is determined. Thus, the dynamical picture is completely established when t = dt.This process can now be repeated to determine the response when t = 2dt, and so on. The turn-on tran-sient resulting from the application of a voltage V(t) = V sin wt, is shown in Fig. 11.9.2.

Note that even though the transient has a well defined wave front, determined by the characteristicline passing through the origin, the characteristic lines are distorted even ahead of this wave front.This is because the applied voltage and the space charge between the screens have an instantaneous effecton the velocity of electrons throughout. Where the characteristic lines converge, abrupt changes in den-sity occur. By increasing the driving voltage, characteristic lines can be made to cross. Electronsentering at one time are overtaken by those entering at a later time. It is to handle this situationthat Lagrangian coordinates are often used. 1

Once an electron has entered the interaction region, so that its initial conditions are established,Oits evolution in the state space (e,n) is determined. This can be seen by combining Eqs. 9 and 10 so asto eliminate time as the parameter:

(1e+ -n(dn = (1 + n)(e

Given an initial position in the state space (e,n), numerical integration of Eq. 17 results in one of thetrajectories of Fig. 11.9.3. It follows from Eqs. 9 and 10 that as time progresses, the trajectoriesare traced out in the direction indicated by the arrows. Thus, the number density in the neighborhoodof a given electron (moving along a characteristic line) is oscillatory in nature, with a frequencytypified by the plasma frequency, w . For the particular initial conditions of Eqs. 13 and 15, whichpertain along characteristics emanating from the t axis, the trajectories all start from the e axis,

but with an amplitude determined by Eq. 15. The picture is now one of particles acting as nonlinearoscillators translating in the z direction with the velocity v.

The perturbation dynamics are governed by the linearized forms of Eqs. 9 and 10, which combineto show that

1. H. M. Schneider, "Oscillations of an Inhomogeneous Plasma Slab," Ph.D. Thesis, Department ofElectrical Engineering, Massachusetts Institute of Technology, Cambridge, Mass., 1969.

Sec. 11.9 11.26

OFig. 11.9.3. Phase-plane (e-n) trajectories of oscillations of electron beam.

d2n + n 0 (18)

dt2

Thus, on a characteristic crossing the t axis when t = to (where n = 0), (19)

n = A(to) sin (t - to)

Linearized, Eq. 8 can be integrated to express the characteristic line along which Eq. 19 applies:

z = U(t - to) (20)

Found from Eq. 20, to can be substituted into Eq. 19 to obtain

wzn = A(t - ) sin ( ) (21)

where dimensional variables have been reintroduced.

The response is the product of a stationary emvelope having a wavelength.Ar 2wrU/wp and a parttraveling in the z direction with the electron velocity, U. The envelope is stationary in spacebecause every electron oscillator passes the z = 0 plane with n = 0. The amplitude of its oscillationis determined by the initial condition on 8 when it passed the screen at z = 0. Note that in this small-amplitude limit, the phase-plane trajectories of Fig. 11.9.3 are circles with radii much less than one.It follows that to achieve linear dynamics, 8 << w .

P

11.10 Causality and Boundary Conditions: Streaming Hyperbolic Systems

Objectives in this section are: (a) to develop readily visualized prototype models for streaminginteractions; (b) to picture in z-t space the evolution of absolute and convective instabilities andof systems which if driven in the sinusoidal steady state would display evanescent and amplifyingwaves; (c) to use the method of characteristics to illustrate the crucial role of causality in thechoice of boundary conditions. In terms of complex waves (and eigenmodes) a small-amplitude versionof the dynamics will be considered again in Sec. 11.12. There, causal boundary conditions, as dis-cussed here, will be essential to understanding the stability of systems of finite extent in the longi-tudinal direction.

Secs. 11.9 & 11.1011.27

Emphasized in this section is the dependence on the longitudinal (streaming) direction. Trans-verse dependences, at least in linear systems, are represented by higher order transverse modes.Linearized, the quasi-one-dimensional models now used represent the long-wave "dominant modes" froma complete small-amplitude model. This interrelationship of models, represented by Fig. 4.12.2, isillustrated in the problems.

Quasi-One-Dimensional Single Stream Models: Planar fluid jets are shown in Fig. 11.10.1. Inthe electric version, the sheet jet is perfectly conducting in the sense that charges can relax onthe interface in times short enough to render the interfaces equipontials. (Perhaps a jet of waterin air.) The jet has a thickness A << a and each of the interfaces has a surface tension y. Elec-trodes to either side of the jet have a potential Vo E aEo relative to the jet.

a Eo

a jEo Ho _

I 0-.oo-L·

(a) (b)

Fig. 11.10.1. Prototype single-stream systems consisting of perfectly con-ducting sheets convecting to the right with velocity U. (a) Poten-tial constrained EQS configuration; (b) flux constrained MQS con-figuration.

For long-wave motions, the transverse electric surface force density, T(z,t), can be approxi-mated by picturing the jet as having a deflection C(z,t) from the center line, with essentiallynegligible slope. Thus, perhaps by using the stress tensor on a control volume enclosing a sectionof the jet, it follows that

S (aE2) (aE )2T ( E 0 2 (1)

(a- ) (a++)

In the magnetic version, the jet is also perfectly conducting, hut now so much so that the mag-netic diffusion time voAa >> 1. The s stem is then the antidual (Sec. 8.5) of the electric one, andT obtained from Eq. 1 by replacing EEo• -,iH2. In either system, the inertial and surface tensionforces acting on the sheet are now also written with the assumption that deflections are slowlyvarying with respect to z. With U defined as the streaming velocity, and approximated here as con-stant, and p the-jet mass density, it follows that Newton's law for motions in the transverse direc-tion is

32zU-) 2 2= +T (2)

The same expression would be written to describe a membrane having surface mass density Ap and tension2y. The velocity of waves on a fixed membrane would then be VE p2

For motions having a typical time scale T, it is convenient to write Eq. 2 in terms of thenormalized variables

( = s/a,t = t/T, z = z/TV (3)

Sec. 11.10 11.28

New variables are introduced:

v = ; e = (4)

so that Eqs. 1 and 2 can be written as two first-order expressions:

ev ev Pe iee P i (5)(- + M +M M(~ + M t) a 1 2 (

ll) ;) (++ M(]

v e = 0 (6)5z _t

where 2 eE2 211H2

o2 2 22 00H2SpAa- = (T/TE) 2or- pa = (Tr/TI)2 and M U/Vpha El pna MI

The last expression follows from taking cross-derivatives of Eqs. 4. Note that P is the square ofthe ratio of the characteristic time to an electro or magneto inertial time, while M is a Mach number.The magnetic and electric systems are respectively described with P positive and negative. With P>O,the transverse force acts in the same direction as the displacement, and hence promotes instability.With P<O, the force acts as a nonlinear spring to recenter the sheet.

Single Stream Characteristics: The characteristic represtntation of Eqs. 5 and 6 follows fromwriting Eqs. 5 and 6 in the form of Eq. 11.6.7 and using the procedures outlined in Sec. 11.6:

dv + de(M + 1) = )2- (1 dt (7)

ond= M + 1; (C-) (8)dt --

It follows from the definition of e, Eq. 4, that

E = e dz (9)

where the lower limit of integration is selected as one where ( is either known or can be related toother variables through a boundary condition.

Because the nonlinearity is confined to the second characteristic equations, Eqs. 8 can beintegrated:

+ +Z- = (M + 1)t + Z- (10)

Thus, the characteristics are straight lines in the z-t plane, as illustrated in Fig. 11.10.2. Bycontrast with the situation in Sec. 11.7, where the second characteristics could be integrated, butthe first not, here the z-t lines along which Eqs. 7 apply are known. It is the second character-istic equations that cause the trouble.

C

QBA,

^4

C

I----S

Fig. 11.10.2

Characteristic lines in z-t plane usedto determine response at C given initialconditions at A and B.

t

Sec. 11.10

--IAI -

3

1/1ýý

c-

i

I

I 1-

11.29

There are two rewards for following the discussion now undertaken of how the characteristics canbe used to give a numerical picture of the dynamics. The finite difference algorithm can be used tocompute the response to initial and boundary conditions in a straightforward fashion. Perhaps moreimportant, the implications of causality for boundary conditions becomes evident in the process.

Consider the determination of the response at C in Fig. 11.10.2, given that at B and A aninstant, At, earlier. With the understanding that Av• and Av! are incremental quantities computedrespectively at the points A and B:

+VA A

VC =A (11)

vB +AB

These two expressions must result in the same response at C. Hence, they can be simultaneously solved.The result is the first of the following four relations between the incremental variables evaluatedat one or the other of the previous points on the incident characteristics;

1 -1 0 0

0 0 1 -1

1 0 M-1 0

0 1 0 M+l

+AvA

A+AeAAe

- B.

vB - VABA

PfAAt

PfBAtB

(12)

where

f 4 )2 2(1 (1 + )

The second of these equations is analogous to the first with v replaced by e. The third and fourthrepresent the second characteristics, Eqs. 7. Solution of Eqs. 12 results in expressions for the in-cremental quantities in terms of the variables evaluated at the previous time step:

Av [VAB(M-I) + [eA-eB](M 2 -1) + .P[fA(M I) - f(M-l)]At (13)

+ + (1)Ae 2 A2 B[VA-vB]"+ (M+1)[eA-eB] + P[fAfB) At) (14)

As indicated by the superscripts, these are the incremental changes in v and e along the C+ character-istics.

Single Stream Initial Value Problem: Suppose that when t = 0, C(z,0) [and hence e(z,0)] andv(z,O) are given at equally spaced points along the z axis. Further, suppose it is decided that forconvenience the response is to be found when t = At at points C similarly selected to fall at inter-vals Az. The values of e, E and v at A and B can be determined from the initial conditions by inter-polating between the initial values.

Then the values of eC and vC, e and v when t = At, follow from Eqs. 13 and 14 used with expres-sions of the form of Eq. 11. Numerical integration, as called for by Eq. 9, then gives the distribu-tion of ý at this time. The situation when t = At is now the same as was the initial one, so theprocess can be repeated to find the response when t = 2At. Thus, the dynamics are unraveled by"marching" forward in time along the characteristic lines. Of course some error will be introducedby the interpolation required to evaluate v and e at A and B and by the numerical integration of Eq. 9.

Typical responses are shown in Fig. 11.10.3. In the absence of a field (P=0) the initial pulsedivides into components propagating upstream and downstream relative to the convecting sheet. Thesepulses propagate without distortion, leaving a null response between. Because they can be representedanalytically, this case gives a check on the numerical scheme (Prob. 11.10.1).

Regardless of the sign of P, one effect of the inhomogeneity is to fill the region between thesepulses with a response. With P < 0, physically the sheet is subject to a spring-like magneticrestoring force. In the extreme of no tension (V = 0), the situation would be one of convecting non-linear oscillators, similar to that considered in Sec. 11.9. The tension adds wave propagation effectsalready familiar from part (a) of Fig. 11.10.3. The combined result, illustrated in part (b) of thefigure, once again shows waves propagating along the characteristic lines, but now attenuating and

Sec. 11.10 11.30

(n1

H H H

UI

U'-F

.U

t3

1•

(b)

U

.U'-f

U

t~

1•

1

(c)

(d)

Fig. 11.10.3.

Typical single-stream initial value responses determined by

numerical integration with Az = .005 and At = .005.

The initial

velocity is zero and displacement is as shown.

(a) With no field, and hence no inhomogeneity, the initial pulse divides into fast

Sand

slow pulses that propagate without distortion.

(b) With P < 0 (magnetic field), wave evanescence results.

For the case shown,

M > 1, so the response is swept downstream.

(c) For P > 0 (electric field) and M > 1 ("sub"), an absolute (nonconvective) instability

results.

(d) For P > 0 but M > 1 ("super"), the instability is convective and the response at a given position remains bounded.

O)

__

leaving behind an oscillating remnant. This oscillating part tends to be carried by the convectionand have an angular frequency -~T

As would be expected for P > 0, which represents a transverse electric force acting much as a non-

linear "negative" spring force, the response in parts (c) and (d) of Fig. 11.10.3 grows with time.T~%.btypes of instability are illustrated. For M < 1, where the flow is "sub" relative to the wavevelocity V, the response becomes unbounded for an observer having a fixed location along the z axis.This is termed an absolute instability or, to distinguish it from the type of response shown forM > 1, a nonconvective instability.

For the convective instability of part (d), M > 1 and the response at a given location remainsbounded. But, for an observer moving downstream it grows. Such an instability can be excited by atemporarily periodic signal at some location along the z axis and a sinusoidal steady-state established

downstream in which the response takes the form of a spatially amplifying wave. At least for linear

systems, such waves are best considered in the frequency domain, as illustrated in Secs. 11.11-11.13.

The nonlinear field coupling has its most pronounced effect in the electric field case. As E - a

(its maximum possible value), the electric force becomes infinite. Thus, the peaks of the deflectiontend to sharpen. In the P > 0 examples shown by Fig. 11.10.3, the initial deflection, consisting ofa cosinusoid plus a constant in the intervals shown, tends to become a triangular pulse.

Quasi-One-Dimensional Two-Stream Models: Consider now the two-stream configurations ofFig. 11.10.4. The sheets have the respective convective velocities U1 and U2 and the same wave

velocities V. They are now not only subject to the "self-field" effects resulting from the electric

and magnetic fields, much as for the single streams, but they are also coupled to each other by thisfield. Thus, a given sheet is subject to "self" and "mutual" forces, represented on the right in the

transverse force equations:

82-lp+2y =1 E- 2fy (15)a D 2 32 1 _ E2 f

Ap(- +U 2 -)2 - 2y 2 2 oo (16)

where, with the displacements normalized to a,

fl(ýl,i2) = I- - (17)1 2 4 + 2 - 2

2 12 4._+C 2 + (18)

H

H_

(a) (b)

Fig. 11.10.4. Prototype two-stream configurations. (a) Potentialconstrained EQS configuration. (b) Flux constrained MQSconfiguration.

Sec. 11.10 11.32

I

With variables defined as in Eq. 4 and normalized as suggested by the single-stream model, these equa-tions are written as four first-order expressions:

a a a a ae-+- = Pfv(+ M

(19)

a a a a De2( + M 2 + M2 + 2 )e2 ae Pf2(2 ,2 2

av ae1 1az at 0

av2 Be22 2at 0Bz Bt =0

(20)

(21)

(22)

Again, for the EQS system, P > 0 while for the MQS system, P < 0.

Two-Stream Characteristics: The same determinant approach used to find the single-streamcharacteristics can be applied to Eqs. 19-22. However, it is more convenient to recognize that theonly coupling between streams is through the inhomogeneous terms. Thus, in view of Eqs. 7 and 8found for a single stream, the characteristics are just what they would be for the individual streamswith the inhomogeneous terms appropriately altered. Thus

dv1 + del(M1 T 1) = Pfl(F 1, 2 )dt

dz

(23)

(24)

(25)

(26)

dv2 + de2 (M2 + 1) = Pf2(1l,)2)dt

dz _ +

dt = M22 1; (+)

The solution at some position, E, when t = t + At is now determined by the response at posi-tions A,B,C and D on the respective characteristics when t = 1, as illustrated in Fig. 11.10.5.:

t t+at t(a) (b)

Fig. 11.10.5. Characteristics in the z-t plane illustrating (a) "super" counter-streaming and (b) "super" stream-structure interactions.

Sec. 11.1011.33

. -

Just as Eqs. 13 and 14 follow from Eqs. 7, Eqs. 23 imply that the changes in vl and el along the C+characteristic from A to E are given by 1

AvlA= 5 (VlVlB) (M-1)+(elA-elB)(M21-1)+P[flA(M+1)- 1B(Q-l)]At} (27)

+ 1AelA = - 1(vlA-vlB)+(elA-elB) (M+l)+P(flAflB)At (28)

Similarly, from Eqs. 25, changes in v2 and e2 along C2 from C to E are as given by these equationswith 1 + 2, A + C and B + D. As before, numerical integration of el and e2 gives the distributionsof 51 and E2. The lower limits of integration in Eq. 9 should be made consistent with the entranceconditions on the respective streams.

Two-Stream Initial Value Problem: Given the initial distributions of ,1'vl, C2 and v2 , theevolution of these variables with time can be determined numerically, much as for the single streams.Given that P can be positive or negative (the two configurations of Fig. 11.10.4) and that M1 and M2can be greater or less than unity (each stream can be "super" or "sub") and can be negative or positive(streaming in either direction), it is clear that there are now many physical interactions that mightbe considered. The super counter-streaming and super stream-structure interactions illustrated inFig. 11.10.5 perhaps add the most physical insight.

The characteristics alone make it clear that with counter-streaming "super" streams it ispossible to have an absolute instability. With P > 0, this instability has much the same characteras for the single "sub" stream. But what might be surprising is the instability that results evenwith P < 0. In this case of magnetic field coupling, the effect of the field on the single streamis to produce decaying oscillations. With two counter-streams,oscillations are fed from one streamto the other and then back to the point of origin with a phase shift. Thus, certain oscillationsbuild up, as the numerically computed response of Fig. 11.10.6 illustrates. Note that the in-stability is unbounded at a fixed position along the z axis. It takes the form of an absolute in-stability, in that the displacement at a given z tends to grow with time.

/

Fig. 11.10.6. Counter-streaming interactions between "super" streams coupled by magneticfield (P = -1000, M1 = 1.5, M2 = -1.5). Two-stream instability is in this caseabsolute.

Sec. 11.10 11.34

Causality and Boundary Conditions: Any real system is of course bounded in the axial direction.One merit of the characteristic viewpoint is that causality is implicit to a specification of theconditions imposed to account for these boundaries. The dynamics unfold along the characteristic lines,always proceeding forward in time. Boundary conditions must be consistent with this requirement.

Consider first the boundary conditions for the single-stream configurations. The differentialequation of motion is second-order in the longitudinal coordinate, so two conditions are required.Mathematically, these could be both imposed at z = 0, both at a downstream position z = L, or one ateach position. But which of these is consistent with having a causal relation between the response andthe initial conditions depends on whether the stream is "super" or "sub."

If the stream is supercritical, both families of characteristics are directed downstream, asillustrated in Fig. 11.10.7a. As time goes on, the response at C that depends on the initial condi-tions between A and B becomes one at C' that depends on both initial conditions and conditions at theupstream boundary. Finally, at points such as C", the response is fully determined by the boundaryconditions at the entrance. Periodic entrance conditions clearly result in a temporally periodicreponse. The supercritical boundary conditions are equivalent to initial conditions and the responseis found following the same line of reasoning as illustrated for the initial value problems. Twoboundary conditions must be imposed at the upstream boundary, but none are imposed on the region0 < z < k by the downstream boundary.

I

E

A

O at t(a) (b)

Fig. 11.10.7. Boundary conditions consistent with causality. (a) Supercriticalcharacteristics implying two upstream conditions and none downstream,(b) Subcritical flow with one condition at each extreme.

By contrast, if the flow is subcritical, as illustrated by Fig. 11.10.7b, two conditions ateither boundary leave the representation over-specified, and one condition must be imposed at eachboundary. To see this, consider how the solution in the neighborhood of the downstream boundary wouldbe found using the characteristics. To find the solution when t = At, the procedure is as alreadyoutlined except for the end points, like C of Fig. 11.10.7b. At this boundary point, v v=C isstipulated (say). Thus, because vA is also known from the initial conditions, the change in v alongthe C+ characteristic incident on the boundary, Av , follows from Eq. lla, From the second character-istic equation along C+, Eq. 12c, the value of Ael is then determined and hence e at the boundary, eC,is found. Thus, e cannot be independently specified as a boundary condition. With the variablesdetermined at the boundaries in this fashion, the stage is set for repeating the process to determinethe solution when t = 2At.

Of course, two boundary conditions can be arbitrarily imposed, say two upstream conditions in asubcritical flow. But it is clear that the resulting solution answers the question, what initial con-ditions are required to make the solution satisfy the desired subsequent conditions at the boundaries?Boundary conditions are usually intended as statements made in advance to predict future events. Ifnot causal, they place requirements on what must have taken place before to have certain conditions atthe boundaries now.

Sec. 11.10.11.35

Consider now causal boundary conditions for the two-stream configurations. The system is nowfourth-order in z and therefore four boundary conditions are required. With the understanding thatcharacteristics have a direction determined by increasing time, one condition must be imposed whereeach family of lines enters the region of interest.

As+an example, consider again ýhe supercritical counter-streaming configuration. The character-istics C1 enter at z = 0 while the C2 characteristics enter at z = Z. To see that the imposition ofthese conditions is consistent with marching forward in time, consider how the solution is determinedwhen t = t + At, given the solution when t = t. Provided that Az and At are selected so that thecharacteristics passing through every interior point on the grid when t = t + At pass through the linet = t in the interval 0 < z < Z, the solution at each of the interior points is found by the sameprocedure as for the initial value problem. The solution at an end point, such as E in Fig. 11.10.8a,is then found by using Eqs. 27 and 28 to find vl and el at E. The boundary conditions provide thevalues of v2 and e2 at E. In this way, the response is determined over the entire interval, includingthe end points, and the stage is set for the next time step. The example of Fig. 11.10.6 is in factcomputed taking into account boundary conditions Cl(0,t), vl(O,t), C2 (£,t), v2 (k,t) all zero. However,time has not progressed far enough to make these conditions significant.

.b..

B

A

S2 =U

B

C

A

t t+nt t t+ At(a) (b)

Fig. 11.10.8. (a) Downstream boundary of counter-streaming configuration atwhich two conditions, on E2 (and hence v2) and e2 , are imposed.

It is because coupling between characteristics for the two streams occurs only through the in-homogeneous terms that this simple procedure takes into account boundary conditions on the counter-steaming supercritical streams.

In Fig. 11.10.8b, the stream-structure interaction makes more evident what is in generalrequired. Stream (1) is supercritical while (2) is not only subcritical but is not streaming at all.To find the response at a downstream boundary, like point E of Fig. 11.10.8b, Eqs. 27 and 28 againprovide (vl, el) at E. One boundary condition is imposed, say v2 is given. Because v2 at C is alsoknown, Av+follows. In turn, Eq. 25a (the second characteristic equation on C2) can be used to solvefor Ae2C. Thus, e2E is determined and all conditions at E are known.

At the upstream boundary, v1 and el are imposed as is a third condition, say that v2 is known.From this last condition the second characteristic equation along C can be used to determine e2 atE'. Again, all conditions at the end points of the grid when t = t + At are established and thestage is set for the next iteration.

In the absence of longitudinal boundary effects, the "super-sub" streaming interaction with-P > 0 is convectively unstable. That is, the response to initial conditions at a fixed position isbounded in time. In this case, boundary conditions have a profound effect. Those just describedturn the convective instability into an absolute one that builds up in an oscillatory fashion.

Sec. 11.10

I h

11.36

LINEAR DYNAMICS IN TERMS OF COMPLEX WAVES

11.11 Second Order Complex Waves

The remaining sections in this chapter continue a subject begun in Sec. 5.17. There, Fouriertransforms are used to represent spatial transients in terms of the sinusoidal steady-state spatialmodes. The example considered there, of charge relaxation on a moving sheet, is typical of a wideclass of linear systems that are uniform in the longitudinal direction, z, and excited from transverseboundaries, perhaps over an interval 0 < z < R. In Sec. 5.17, the resulting temporal sinusoidal steadystate consists of responses, shown in Fig. 5.17.8, that spatially decay upstream and downstream fromthis range. These are a superposition of the appropriate spatial modes. Within the excitation range,the response is also a superposition of spatial modes. But in addition, in this range there is thedriven response having not only the same temporal frequency as the drive, but the same wavenumber as

well. The Fourier transform provides a formalism for "splicing" the modes and driven response together

in the planes z = 0 and z = R.

As pointed out in Sec. 5.17, there are two questions left unanswered in the process of findingthe spatial transient. First, it is assumed there that the sinusoidal steady-state complex wavesdecay away from the excitation region. Thus, those spatial modes having positive imaginary k are ex-cluded from the downstream range k < z, while those with negative imaginary parts are left out in theregion z < 0. The examples introduced in this section include the possibility that the downstreamresponse in fact grows with increasing z. In Sec. 11.12, the objective is to have a means of dis-tinguishing such amplifying waves from those that are evanescent, or decay away from the drive.

Any discussion of a sinusoidal steady state is predicated on having an answer to the secondquestion. Is the system absolutely stable, in the sense that the response is bounded with increasingtime at a fixed location in space? Only then will the temporal sinusoidal steady state have a chanceto establish itself. The difficulty here is that a system is not necessarily absolutely unstableeven if it displays temporal modes with negative imaginary frequencies. Temporal modes, having fre-quencies given by the dispersion equation evaluated using real values of the wavenumber, are theresponse to initial conditions that are spatially periodic. These extend from z = -- to z = +-o.Thus, in an infinite system, temporal modes do not suggest whether the instability grows with time ata fixed location, z (absolute instability), or rather grows only -for an observer that moves with somevelocity in the z direction (convective).

The identification of an absolute instability is taken up in Sec. 11.13.

Second Order Long-Wave Models: It is the purpose of this section to set the stage for the nexttwo sections. Although the w-k picture of the evolution of a system in space and time is widelyapplicable, it is helpful to have in mind simple situations that make it possible to establish aphysical rapport for what the mathematics represents. In Fig. 11.11.1, sheets of liquid stream inthe z direction between plane-parallel plates or electrodes. (These same configurations are con-sidered from another point of view in Sec. 11.10.) With the stream in steady equilibrium there is notransverse deflection, E, and there are uniform fields between the plane-parallel perfectly conductingwalls and the sheets, as shown. Over the range 0 < z < k, the transverse boundaries are driven eitherby electric or magnetic potentials superimposed on the uniform bias potentials which give rise to theequilibrium fields.

The models now developed highlight the dominant modes of systems actually having an infinitenumber of spatial modes. The higher order modes that are left out of the models come into play if"wavelengths" of interest are as short as the spacing, a, or the sheet thickness, A. The magneticconfiguration is the antidual of the electric one, as defined in Sec. 8.5. Thus, the equation of mo-tion follows directly from the electric case now derived, with Eo + o, Eo + Ho, 4 -+ A/po.

With the stream modeled as a membrane having a tension twice that due to the surface tension,the equation of motion is Eq. 11.10.2. Taking into account the excitation potentials, the transversesurface force density in the long-wave limit is simply the difference between the electric stressesacting on the top and the bottom of the sheet:

1 (-Eoa + d 2 (-Ea-$d) 2 (1)T =o d oo jd

(a - ) (a + )

For small amplitudes of the drive and response,

2s E 2c E2

T 00 o (2)*a d a

This expression is now used to complete the transverse force equation, Eq. 11.10.2, which takes the

11.37 Sec. 11.11

/cD

E~a

(D,(z

,t)

z~~

t

a E.

6

/A=

-ýL.

HHa

-/(

z,t)

A(z

Ht)

z

H,

"A=-

ýL.H

Ha

-A,(z

,t)

(a)

P>O

, M

< I

.. .....

.... .

.. ..

. ...

.. .

w(b

)

k

(d)

Fig. 11.11.1.

Electric and magnetic long-wave models that are described by Eq. 3 together with dispersion relations.

Complex values of k are shown for real values of w: kr -

, ki

----

. (a) Subcritical electric, (b) sub-

critical magnetic, (c) supercritical electric, and (d) supercritical magnetic.

1;·;;;

normalized form (5 g=a, t = tT, and z = zTV):

(- + M -)2 ~ + PE - Pf(z,t)

The membrane wave velocity and Mach number are defined as

V = P; M= U

and "pressure" parameters and forcing functions making the equation applicable to the electric and mag-netic configuration, respectively, are

2 E2T2p= ooP 00

Apa

f- aEo

21oH T2

ApaAm

f = aH-- oaHo

Thus, with P > 0 the configuration is electric, and the self-field part ofstabilizing. That is, a deflection results in a field intensification andon the stream tending to further increase the deflection. In the magneticand it is as though there were a continuum of magnetic springs between theflection leads to a force tending to return the stream to its equilibrium.

the transverse force is de-hence a transverse forcefield configuration, P <stream and the walls. A

Whether the underlying method of characteristics from Sec. 11.10 has been followed or not, it isuseful at this point to review the response to initial conditions, shown in Fig. 11.10.3, for thesestreaming configurations. It is the Mach number, M, that determines if the initial pulse can propa-gate upstream. For M < 1, wave fronts propagate in both directions, whereas if M > 1, the entire re-sponse is washed downstream.

If P is positive, the amplitude becomes unbounded. But, whether the growth is at a fixed loca-tion or for an observer moving with some velocity depends on M. Thus, in this electric case, the in-finite system is absolutely unstable if M < 1 and convectively unstable if M > 1.

If P is negative, the response consists of forward and backward waves. The magnetic fieldresults in their leaving an oscillation in the region between. If M < 1, this oscillation decayswith time at a fixed location, while if M > 1, the response falls abruptly to zero as the wave frontthat is trying to propagate upstream is swept downstream.

With these predispositions as to whatshould be expected, consider now the repre-sentation of the dynamics in terms of com-plex waves.

Spatial Modes: Consider first the re-sponse to excitations that are in the sinu-soidal steady state, having a frequencyw = bo. Because they involve the same mani-pulations, but contrasting issues, two typesof problems are now considered. In the first,the system is bounded by the planes z = 0 andz = Z. The transverse boundaries are notdriven. Rather, the drive is through one ofthe longitudinal boundary conditions whichvaries at the angular frequency wo.

The second type of problem is one ex-tending from z = -- to z = +oo with the exci-tation from the transverse boundaries overthe range 0 < z < £. These bounded and un-bounded situations, pictured in Fig. 11.11.2,are similar enough that they are now con-sidered at the same time.

To be consistent with normalization ofEq. 3, deflections are taken to be of the form

inhomogeneous boundaryconditions at frequency w0

homogeneous

/boundary conditions

1-- - - - - - - -- .--3

n1

(b)

Fig. 11.11.2. Typical systems that are uniform inthe z direction; (a) bounded longitudinallyby planes where boundary conditions are im-posed; (b) unbounded in the z direction withdrive from a transverse boundary over theinterval 0 < z < k.

=Re ~e(wt-kz)

Sec. 11.1111,39

where frequencies and wavenumbers are normalized such that wT = w, kTV = k. The inhomogeneous solutionto Eq. 3, caused by the drive on the transverse boundaries and applying over the range 0 <,z < iJ,fol-lows by substituting this form of solution into Eq. 3 with w = wo and k = (. Solving for ý gives

Pf=D(w,1) (7)

where the dispersion function is

D(w,0) = (Wo - M )2 _ 2 + P (8)

For solutions of the form of Eq. 6 to satisfy the homogeneous form of Eq. 3, k must satisfy the dis-persion equation D(w,k) = 0. Thus, with amplitudes A and B at this point arbitrary, the solution overany range of z is

[Pie-jBz + -jklz + -jk 2 zI jW 0 tS= ReD--(o~ + Ae + Be Je (9)

where

D(w,k) = ( - Mk)2 - k2 + P (10)

Thus, the wavenumbers k1 and k2 in Eq./8 are given by solving the quadratic expression D(wo,k) = 0:

k1 = Tj + (11)-1

where

woM W•2 + P(l -M 2)

M2 - 1 M2 - 1

As a graphical representation of the spatial modes, these roots are displayed in Fig. 11.11.1.For a particular driving frequency wo, the roots of D(wo,k) are represented by the intersections withthe horizontal line. The solid curves indicate the real part, kr, while the broken lines are theimaginary part, ki . Where one k is shown, it is in common to both roots.

Four possibilities are distinguished in Fig. 11.11.1. The configuration can be electric or mag-netic (P > 0 or P < 0) and it can be subcritical or supercritical (IMI < or IMI > 1). As can be seenfrom Eq. 1, two of the four have ranges of frequency over which the wavenumbers are complex:

S2< P(M2 1) (12)o

One is the subcritical magnetic case, where P < 0 and IMI < 1. The other is the supercritical elec-tric case, where P > 0 but IMI > 1. In each of these, one spatial mode apparently "grows" with in-creasing z while the other "decays." Of course, in the magnetic subcritical case the spatial modethat appears to grow in the z direction is really an evanescent mode decaying upstream from a down-stream drive. The supercritical electric case actually does involve a wave that is amplifying in thez direction as it moves away from an upstream source.

Section 11.12 shows how the distinction can be made between evanescent and amplifying waves byconsidering how the waves are established in the sinusoidal steady state subsequent to turning on theexcitation.

The remainder of this section is intended to develop a physical understanding of evanescent andamplifying waves and of absolute and convective instabilities.

Driven Response of Bounded System: If IMI < 1, boundary conditions can be imposed at z = 0 andz = k. That these conditions are consistent with causality can be established by the method of char-acteristics (Sec. 11.10), or by using the arguments of the next section to determine that one spatialmode propagates in the +z direction (and hence can be used to satisfy a boundary condition at z = 0),while the other propagates in the -z direction (and can be used to satisfy the condition at z = £).As an example, suppose that the sheet is given a sinusoidally varying excitation at z = 0 and fixedat z = £. Also, make the transverse boundary excitation zero, so f = 0. Then, the coefficients Aand B in Eq. 9 are determined and the solution becomes: (Note that in the normalized expression,

j=£/tV.)

-Re sin y(z - ) t- (13)d sin y2

Sec. 11.11 11.40

If the frequency is below the cutoff frequency, Eq. 12, y is imaginary and the evanescent nature of

the response is made more evident by writing Eq. 13 as

sinh y(z - ) J(Wot-nz)f= -Re d sinh ya e (14

Some features of this steady-state response are illustrated by the experiment shown inFig. 11.11.3. Here, the sheet is replaced by a wire under tension. In the absence of a magneticforce, it too has a deflection described by the wave equation. There is no longitudinal motion, soM = 0 and hence r = 0 in Eqs. 13 and 14. By passing a current through the wire and imposing a mag-netic field that is all gradient along its zero-deflection axis, a magnetic force is produced that isproportional to F. This force tends to restore the undeflected wire to its original position. Theconfiguration is described in Prob. 11.11.2, where it is shown that the equation of motion is againEq. 3 with P < 0 and M = 0.

In the first picture of the sequence, the current is zero and what is seen is the standingwave resulting from the interference of two oppositely propagating ordinary waves. (In these pic-tures, the z direction is to the left, so the excitation is to the right.) The frequency is suchthat the wire is very nearly in the lowest resonance condition that prevails if yZ = nff. As thecurrent is raised, the magnetic force tends to counteract the inertial force (that makes the wirebow outward). The current is reached where these forces just balance, and the deflections decay awayfrom the excitation. The rate of decay is largest at zero frequency (a static deflection).

Consider next the dramatic effect of having the continuum not only stream, but be supercritical,so that IMI > 1. Then, two boundary conditions must be imposed at the inlet, where z = 0, and nonethat influence matters in the range of interest are imposed at the exit. For example, the deflec-tion is again the sinusoidal one assumed before, but the spatial derivative is constrained to bezero. Then, the coefficients A and 9 are determined and the solution is

(klejYz - k e- jY z ) j(wot-nz)=Re Ed 2y

The case of most interest has the electrip configuration of Fig. 11.11.1 as a prototype and henceP > 0. If P is raised high enough that < P(HI1 - 1), y is imaginary, and the space-time picture ofthe deflections given by Eq. 15 is more apparent if it is written in the form

S(tlellz - k l e - YIz) j(.ot-fz)S= -Re Ed 2y e (16)

In the case of this supercritical stream, a demonstration is made by letting the continuum be ajet of water, with capillarity providing the (surface) tension (see Prob. 11.11.3). The drive is pro-vided by spherical electrodes positioned just upstream of (z = 0) on each side of the stream and bi-ased by a constant potential relative to the stream with a superimposed sinusoidally varying voltagehaving the angular frequency w.

With P = 0, so that y is real, the response is illustrated in Fig. 11.11.4. (Again, streamingis from right to left with the excitation at the right.) The fast and slow waves carried downstreamby the convection interfere to form "beats." That is, the envelope of the deflection is a standingwave having wavelength 27/y. In Fig. 11.11.4b, the frequency has been raised to the point where aboutone half-wavelength of the envelope appears within view. In a slow motion picture (Complex Waves II,Reference 11, Appendix C), the phases propagate through this envelope with velocity wo/h.

With a field applied to the jet, the kinking motions of the jet are very similar to those of theplanar sheet. Thus, raising the voltage is equivalent to raising P, and has the effect on the dis-persion equation and jet that is illustrated in Fig. 11.11.5. Over most of its length, the responsedescribed by Eq. 16, is dominated by the growing exponential. Again phases have the velocity wo/with the exponentially growing envelope.

Instability of Bounded Systems: The importance of imposing boundary conditions that are consist-ent with causality is made dramatically evident by considering implications for stability of correctand incorrect conditions. For a bounded system, it is not meaningful to envision a convective insta-bility. Once boundary conditions have been imposed, there remains only the possibility of an abso-lute instability in the response to initial conditions. This transient response is representedby aRuperposition of modes satisfying homogeneous transverse and longitudinal boundary conditions (f and

+d- 0).

For the subcritical system, it can be seen from Eq. 13 that the eigenvalues for these modes are

Sec. 11.1111.41

en

(1)

(l f-'

f-'

f-'

f-'

( a

) (b

)

f-'

f-'

-I'­

N

(c)

( d

)

Fig

. 1

1.1

1.3

. T

ime

ex

po

sure

o

f sin

uso

idal

ste

ad

y sta

te fo

r a

"str

ing

"

rep

resen

ted

b

y

Eq

. 3

wit

h M

=

0,

P <

0

an

d

the ex

cit

ati

on

at

the

rig

ht.

(a

) M

ag

neti

c fi

eld

is

o

ff

(P

=

0)

an

d

freq

uen

cy

is

su

ch

th

at

resp

on

se is

n

ear

the fir

st

reso

nan

ce.

(b)

Mag

neti

c fie

ld

rais

ed

so

mew

hat

, b

ut

freq

uen

cy

still

ab

ov

e cu

toff

. (c

) F

ield

ra

ised

to

cu

toff

co

nd

itio

n.

(d)

Ev

an

esc

en

t re

spo

nse

re

su

ltin

g

as

field

is

ra

ised

to

a

po

int

wh

ere

fr

eq

uen

cy

is

le

ss

than

cu

toff

. (F

rom

C

om

ple

x W

aves

I,

Refe

ren

ce

11

, A

pp

en

dix

C

.)

-I'­

N

Courtesy of Education Development Center, Inc. Used with permission.

( a )

Fig. 11.11. 4

Supercritical stream (M > 1) with no field (P = 0) and excitation to the right. Raising the fre­quency just brings one half-wave­length of "beat" into view. (From Complex Waves II, Refer­ence 11, Appendix C.)

given simply by

sin '(9., = 0 ~ '( = nTf/9."

( b)

(17)

In evaluating this expression, using the definition of '( given by Eq. 11, the frequency is now the eigenfrequency, conveniently represented here as jW ~ sn. Thus,

s2 = _(nTf) 2 (M2 _ 1) 2 + P (1 _ M2) (18)n 9., ~

Because IMI < 1, it follows that P must be positive if there is to be instability. As P is raised, the n = 1 mode is the first to become unstable and that occurs if

P (19)

At this threshold, the deflection has a shape given by Eq. 13, with an envelope having the shape of a half-wave of a sinusoid. This instability is illustrated in the limit M ~ 0 in Complex Waves II (Reference 11, Appendix C).

Consider the consequence of an unjustified use of Eq. 18. Suppose that it is valid for the supercritical case, IMI > 1. It would then be concluded that the system is unstable with P made sufficiently negative (the magnetic case in Fig. 11.11.1). Of course, with IMI > 1, one boundary condition underlying the identification of these eigenmodes is not consistent with causality. From

11.43 Sec. 11.11

Courtesy of Education Development Center, Inc. Used with permission.

n = 1,2,3···

C/l

('D

() ......

......

......

......

(a)

(b)

f-'

......

-l'­

-l'­

(c)

(d)

(

Fig

. 1

1.1

1.5

. L

iqu

id je

t in

ele

ctr

ic fi

eld

is

ex

cit

ed

in

th

e s

inu

soid

al

stead

y sta

te b

y

the

sph

ere

s to

th

e ri

gh

t.

Am

pli

fyin

g w

aves

, m

odel

ed b

y

Eq.

3

wit

h

P >

0

and

M >

1.

Part

s (b

),

(c),

an

d

(d)

show

th

e eff

ect

of

rais

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the cu

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Com

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Wav

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11

, A

pp

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C.)

Courtesy of Education Development Center, Inc. Used with permission.

~

the correct solution, Eq. 15, it is clear that in this supercritical case there are no eigenmodes,never mind modes that are unstable.

Driven Response of Unbounded System: Consider now the sinusoidal steady-state response to a drive

from transverse boundaries in the unbounded configuration, Fig. 11.11.2b. For the quasi-one-dimen-sional model, solutions are piecewise continuous in the z direction. In regions I and III there is nodrive and hence f = 0, while in region II there is a drive. With an appropriate assignment of f, thegeneral solution, Eq. 8, can be applied to each region. There are two coefficients, A and B, associ-ated with each region. These represent the amplitudes of the spatial modes and are determined byboundary conditions at infinity and by the conditions prevailing where the regions meet at z = 0 andz = 2.

A picture of the sinusoidal steady-state response in the four regimes illustrated in Fig. 11.11.1is given in Fig. 11.11.6. First, consider the subcritical situations. Here, boundary conditions mustexist at z + - and z - -- that have an effect on the asymptotic response. So long as the waves arepropagating (P > 0 and P < 0 but the frequency above cutoff), it is necessary to specify conditionsat infinity. One such specification might be a "radiation condition," which requires that boundariesare far enough removed that waves reflecting their presence have not returned to the region of exci-tation, or that these boundaries absorb the incident wave without there being any reflected wave.In either case, for IMI < 1, the response is

jwot= Re e

-jk 1zJe ; z< 0

.- jBz -jk z -jk zD(,)+ +e e ; 0 < z < (20)D(11) II II

-jkl z

BIIIe <z

where modes representing conditions at infinity have been excluded from regions I and III.

The four coefficients are determined by making the displacement and its spatial derivative piece-wise continuous. That is, requiring that ý and ac/az be continuous at z = 0 and z = £ gives four con-ditions allowing the four amplitudes to be determined in terms of f.

For P > 0 and for P < 0 with the driving frequency above cutoff, the response outside the exci-tation region consists of purely propagating waves. Thus, the envelope of the response in the ex-terior is constant in z. In the magnetic case, where P < 0, the response below the cutoff frequencyconsists of evanescent waves, as sketched in Fig. 11.11.6. As suggested by the general form of thesolution, Eq. 20, in the excitation region the response is a sum of the spatial modes representingend effects and a driven response that has the same wavenumber as the drive. For operation below thecutoff frequency, the response in the mid-range of region II at distances removed by several decaylengths from the ends would be just the part having the same spatial periodicity as the drive. Thistype of behavior is familiar from Sec. 5.17, and also illustrated in detail by Fig. 5.17.8.

The case P > 0 and IMI < 1 is absolutely unstable. Sooner or later the response to initial con-ditions would dominate the sinusoidal steady state.

Now, consider the effect of having a supercritical stream, IMI > 1. The response in region I isentirely determined by the upstream boundary conditions. If those conditions are homogeneous, or thatboundary is too far upstream to have had an effect during times of interest and initial conditions onthe stream are zero, then the solution in region I is known to be zero. With this understanding, theresponse then takes the form

-=Re e

0; z < 0

pe-jý z -jkiz -jkz

D(w0, ) + Aie + BIIe ; 0 < z < k (21)

-jk z -jk z

AIIIe +BIIIe ; R < z

The response continues to evolve in the direction of streaming. In region II, the amplitudesare fully determined by the requirement that ý and 3E/Dz be zero at z = 0. In turn, the downstreamresponse in region III follows by requiring continuity of E and a3/3z at z = £.

Sec. 11.1111.45

P>O P(O

I BC 2,JC 2.JC I BC I BC 21JC 2JC I BC

M <' I

\traveling waves evanescent or

traveling waves

2BC 2,JC 21JC 2BC 2.JC 2,JCs I I

M I I

amplifying or beating traveling waves

traveling waves

Fig. 11.11.6. Boundary conditions (B.C.) and jump conditions (J.C.) for secondorder unbounded systems.

In the case P > 0 (the unstable configuration), operation below the cutoff frequency results inan amplifying wave. Thus, it is possible that even within the excitation region the spatially periodicresponse will not prevail. The spatially amplifying wave certainly dominates in the downstream region,since the only other contribution to the response in that region is a decaying wave.

The downstream responses for the stable and unstable supercritical cases of Fig. 11.11.6 areillustrated experimentally by Figs. 11.11.4 and 11.11.5, respectively.

In retrospect, what is the intellectual basis for the association of the spatial modes with bound-ary conditions at infinity that made the difference between the supercritical and subcritical solu-tions? (Certainly it is not an identification of the direction of propagation of the phases.) Infact, left at this point, it is necessary to fall back on the method of characteristics, Sec. 11.10, tojustify the association of modes with boundary conditions. In the next section, the objective is tohave a method of relating the modes to the conditions of causality. The excitation will be turned onand the appropriate solution found as the asymptotic response. This approach can be used in systemswhere the method of characteristics is not applicable.

It has been presumed in this discussion of the response for the infinite system that at a givenpoint in space, the response remains bounded as time increases. It will be the purpose of Sec. 11.13to identify conditions for an absolute instability and to discriminate between it and a convective in-stability.

11.12 Distinguishing Amplifying from Evanescent Modes

Whether the excitation is from transverse or longitudinal boundaries, the sinusoidal steady-state asymptotic response is a superposition of waves having complex wavenumbers, k, for a realfrequency, w. To understand how these modes are to be combined in this long-time limit, it is neces-sary to picture the response in relation to the turn-on transient from which it arises.1,2

1. For a more complete exposition of criteria for identifying the types of complex waves in in-finite media, see R. J. Briggs, Electron-Stream Interaction with Plasmas, The M.I.T. Press,Cambridge, Mass., 1964, pp. 8-46. Also, A. Bers, Notes for MIT subject "Electrodynamics ofWaves, Media, and Interactions."

2. A. Scott, Active and Nonlinear Wave Propagation in Electronics, Wiley-Interscience, New York,1970, pp. 27-44.

Secs. 11.11 & 11.12 11.46

the outset that the temporal dependence can be represented by a complex amplitude, for example D =Re$(x,y,z,w)exp(jwt). The spatial transient is in turn represented by a Fourier transform. To permita representation of the transient which joins the initial conditions to a possible sinusoidal steadystate, this temporal complex amplitude is now replaced by the Laplace transform pair

jw t d -jwt(xz,t) = (x,z,w)et 2 $(x,z,) = (x,z,t)e dt (1)

xzt)= )-jo o

As in Sec. 5.17, the longitudinal dependence is in turn represented by the Fourier transform

+co +CO

j-kz dk -jP(x,z,W). (xkw)ekz d (x,k,w) = (xzw)e dz (2)

The Laplace transform, Eq. lb, starts when t = 0, and so the transform of temporal derivatives bringsin initial conditions. For example, the Laplace transform of the first derivative is integrated byparts to give

t -jwt j= De-j t - (-jw)De dt = -D(x,z,O) + jw4 (3)

Thus, if a variable is zero when t = 0, then the Laplace and Fourier transform of its temporal deriva-tive is simply jwP. Of course, the Laplace and Fourier transform of the derivative with respect to zis -jk$. That is, relations between complex amplitudes apply also to Laplace-Fourier transforms, pro-vided that rest conditions prevail when t = 0 for the transformed variables. If there are finiteinitial conditions, then care must be taken to include them in transforming all relations.

As an example, consider the second order systems represented by Eq. 11.11.3 in an unbounded con-figuration. The excitation is from transverse boundaries (Fig. 11.11.2b) where the forcing functionis imposed over the interval 0 < z < R as a traveling wave

j(w t-fz)f = u (t)[u (z) - u (z - k)]Ref e

Sj(wot-Bz) -j( t-ýz)= u(t)[ul(z) - ul(z - )][foe + fo e

1-1 2o o

Substitution, first into Eq. lb and then into Eq. 2b, gives the transform of the forcing function as

[l j(k-ý) ^* j(k+)f[ e ] f[l - e j ( k + ) ]

f 0 + 0 (52(w - w )(k - B) 2(w + w )(k + 3)

Note that the second term is obtained from the first by substituting fo - fo, wo -* -o, and + -8.With the understanding that the real response is the sum of the one now found and a response formed bymaking this substitution, it will be assumed in what follows that the drive is just the first term inEq. 5. (Note that fo is not a transform, but rather simply a complex number expressing the phase andamplitude of the drive.)

With the understanding that when t = 0, C and ý3/3t are zero, the Laplace-Fourier transform ofEq. 11.11.3 gives an expression for the transform of the sheet deflection:

Pf 2 2

D(w,k)' D(w,k) (w - Mk) - k + P (6)D(wk)

This expression is obtained either by treating variables as though complex amplitudes are being intro-duced or by starting from scratch by multiplying Eq. 11.11.3 by exp[-j(wt - kz)] and then integratingboth sides of the expression from 0 to - on t and from -- to o on z. Integrations by parts of theterms involving derivatives and the definitions of the transforms, Eqs. lb and 2b, then also resultin Eq. 6. (Note that consistent with the normalization used in Sec. 11.11 are k H kTV and W = wT.)Thus, in view of Eq. 5, the desired Laplace-Fourier transform is written in terms of the specifictraveling-wave excitation, turned on when t = 0, as

D(w)q(k)D(w,k)

Sec. 11.12

T,,1,,,,,~ a,..,i,,'P~nnnF~~m D~r\rr\r~ntnt-;~n ;n 'T~imd onrl Cn~~o· Tn SnF ~17 i+ ip pcctimp~rl frnm

11.47

where

Po [e -j) k_ 1]g(k) -2 j (k - a)

Sj( - Wo)

The model represented by Eqs. 8 and 9 is long-wave. But, the form of the response transform taken byEq. 7 is representative of a much wider range of physical situations that are uniform in the z direc-tion. The details of a transverse dependence are determined by solving differential equations andboundary conditions over the transverse cross section. This amounts to representing the transformedvariables in terms of trahsfer relations and boundary conditions, as exemplified many times inChaps. 5, 6, 8, 10 and in Sec. 11.5.

Laplace Transform on Time as the Sum of Spatial Modes; Causality: The evolution in z-t spaceis determined by using Eq. 7 in evaluating the inverse transforms, Eqs. 1 and 2:

+4o

(z,) = f g(k) e-jkz dk-z)= 6LOD(w,k) 2T (10)

Fig. 11.12.1. Fourier contour.

( )

C(z,) =

i(z,w)ejwt d~w2z)e2

Fig. 11.12.2. Laplace contour.

The w that appears in Eq. 10 is any one of theThat is, the integral on k is carried out withto subsequently carry out the integration on w.

frequencies on the Laplace contour, CL of Fig. 11.12.2.w each of the (generally complex) frequencies required

Causality is built into the inversion of the transforms through the choice of the Laplace con-tour. On this contour, w = w - jo, where a is constant. Thus, in Eq. 11, exp(jwt) = exp(jwrt)exp(at).Thus, for t < 0, the integran5 goes to zero as a - 0, and the integrand along the Laplace contour canbe replaced by one closed in the lower half plane. That the response for t < 0 must be zero is there-fore equivalent to requiring that this integral over the closed contour C t vanish. The closure isillustrated by Fig. 11.12.3. Cauchy's theorem makes it clear that thij causality condition will pre-vail, provided that the Laplace contour is below all singularities of F(z,w) in the w plane.

For any given frequency on this contour, the situation for inverting the Fourier transform asspecified by Eq. 10 is no different than in Sec. 5.17, except that the frequency is in general somecomplex number. In Sec. 5.17, where it is assumed at the outset that sinusoidal steady-state condi-tions prevail, the frequency is the real frequency of the drive.

Note that g(k) is not singular at k = B. Further, for the second order system, and for othershaving dispersion equations that are polynomial or transcendental in k, the roots of D(w,kn) = 0represent poles in the k plane. Thus, if the integration on the Fourier contour can be converted toone that is closed, then Cauchy's residue theorem

Sec. 11.12

(11)

ýWi

a-

I

11.48

Fig. 11.12.3. Closure of Laplace contour to Fig. 11.12.4. Closures to evaluateidentify CL consistent with causality. Fourier integral.

N(k )

N(k) dk = +2j(K + K2 + ); K = kn) (12)C D(k) n D'(kn

provides a simple evaluation. The positive and negative signs are for counterclockwise and clockwisedirections of integration. For a given frequency on the Laplace contour, the Laplace transform isthe sum over the spatial modes.

To see what closed contour can be used to replace the open one that is to be evaluated, observethat in Eq. 10

[ej ( k -B _ 1][e - j k z ] = e - j e e - e e (13)

If z is in the range z < 0, the entire term goes to zero if ki - +=o. Thus, the Fourier integration canbe replaced by one on the closed contour Cu in the upper half of the k plane, as shown in Fig. 11.12.4.If z is in the range k < z, the entire term goes to zero if the contour is closed in the lower halfplane, on Ck. In the excitation range, where 0 < z < Z, the terms in Eq. 13 must be treated separately.The individual functions are singular at k = B, so the response in this range includes not only thespatial modes, but a "driven response" having the wavenumber B. This is considered in detail inSec. 5.17, and will not be highlighted here. With the understanding that the summation is made appro-priate to the range of z being considered (to left or right of the excitation range), Eq. 10 is in-tegrated to give the Laplace transform

=j9g(kn) -jknz(z,w) = +(() D () k e (14)

nD'(,k)

where the upper and lower signs pertain to the upper and lower contours in Fig. 11.12.4. For example,in the case of the second order systems, where Eq. 6 gives the dispersion equation,

D(w,k) = (M - l)(k- k )(k - kl)

SwM 2 + P(1 - M2k, M+ - (15)

-1

It follows that in Eq. 14

22 -

D'(kl) =+ (M2- l)(k - k_) = + 2 w2 + P(l - M) = 2 1( - cc)(w+ c) (16)

-1

Sec. 11.1211.49

Asymptotic Response in the Sinusoidal Steady State: The Laplace contour, CL, lies below

all singularities of t(z,w). In the second order systems, there are singularities of the individualterms in Eq. 14 at +c." Whether or not they represent singularities of ((z,w) depends on the yet tobe determined appropriate summation in Eq. 14. These branch poles of the individual terms are illus-trated for the submagnetic case in Fig. 11.12.5. They are designated as branch poles because thefunction D'(w), where kn = kn(W), is double-valued if w is allowed to pass through the branch linejoining these poles (see Fig. 11.12.5). Because of J(w), there is clearly a pole of E(z,w) at =wo .

Fig. 11.12.5

A

ii/1 li/I

-Wc

Wi W For second order system, singularities of individual

+WC \ terms of Eq. 14 are branch poles at w = +w , where--WC C

-c /-p(l-M2). Branch lines of (M2-1)(k+-kC )

8_ &+ /(-• )(W+k ) are defined so that the principal valueSc cf oa comlex, vaial A ex~-g~(j6 is - < 6

The objective here is not to carry out the second integration called for with Eq. 11, but ratherto discern the response when t + C. At a given location, z, there are two long-time possibilities.The response can either reach the sinusoidal steady state, or it can become unbounded. To achieve theformer, it must be possible to move the Laplace contour, CL, so that it is as shown in Fig. 11.12.6.This is possible if there are no singularities below the(open) contour. The part, C', runs parallel to the realaxis with wi slightly positive. Thus as t + m, the in-tegrand of Eq. 1i on CL goes to zero, and this part ofthe integration gives no asymptotic contribution. The CLcontributions to the integral along the oppositely di-

ll/ \Wrected segments, L, cancel. nhus, the Integral reducesto a closed integral on C"'. The only singularity with-in this contour is in 6(w, at o = wo. Thus, CL L

g(kn) j(wot-knz) Wo(z,t) = (kn)ej(tk ; k = k (wo) (17)

-n n n 0 Fig. 11.12.6. Laplace contour inlimit t + C with no singu-

where upper and lower signs pertain to closures of thecontour in the upper and lower plane.

As an example, consider the submagnetic second order system (with IMI < 1 and P < 0). In thiscase there are no singularities in the lower half of the w plane, but there is the embarrassment ofbranch poles on the wr axis, as illustrated in Fig. 11.12.5. However, these occur on the axis becausethere is no damping in the system. If a force term is added to the equation of motion (Eq. 11.11.3)having the form

f = -B(ýt + Uvz)( (18)

where B is a positive damping coefficient, the branch poles are displaced into the upper half plane.

It is now possible to identify the proper contributions to the Laplace transform, Eq. 14, andhence to the sinusoidal steady-state response given by Eq. 17. For a given Laplace contour, CL(a a finite positive constant), the poles, kn, form a locus of points in the k plane. Again, for thesubmagnetic case (P < 0 and IMI < 1), one locus is in the lower half plane and the other in the upperhalf plane. As the Laplace contour is displaced upward to the Cr axis, these loci terminate in con-tours of complex k for real w shown as dashed lines in Fig. 11.12.7. More prominently shown in thisfigure are the contours followed by the poles, kn, as the Laplace contour is displaced upward to the

Wr axis. On these contours, wr is constant and a is decreasing to zero. For example, holding Wr =0.8, as a is reduced to zero gives one pole that moves down and to the right in the second quadrant(kn = k-1) and a second that moves upward and to the left starting in the fourth quadrant (kn = kl).As a reaches zero, these poles become complex conjugates. In general, one pole comes from above andone from below, each terminating on the locus of k for real w. In this example, no pole staiting inthe lower half plane reaches the upper half plane and vice versa.

It is now clear that the n = -1 pole constitutes the only term in the sum in Eq. 17 with closureof the Fourier contour in the upper half plane (for z < 0) while n = 1 and closure in the lower halfplane is appropriate for z > 0. Thus, Eq. 17 becomes

Sec. 11.12

B

WC W rvys~r~~vr~~Jl I

11.50

Fig. 11.12.7. Loci of k for loci of w shown by inset. Broken curves arevalues of k for real w. Submagnetic case (IMI < 1, P < 0) displaysevanescent waves (P = 1, M = 0.5).

j(k -3)Z j(wot-k_ z)Pf +1 +1

o [1 - e ]e < 0S(z,t) ; z (19)2 (k - )(k1 - kl)(M - 1)

+1

That is, if the frequency is in the range wo < lWcl where waves are cut off, these waves are evanes-cent. They decay away from the excitation. For example, that k_l has a positive imaginary partdoes not mean that it represents a wave that grows in the +z direction, but rather that it is a wavedecaying in the -z direction. Converted to a real function of time in accordance with the discus-sion following Eq. 5, the result given by Eq. 19 is consistent with the sinusoidal steady-stateresponse deduced in Sec. 11.11 for this case (Eq. 11.11.20).

Consider by contrast the establishment of a sinusoidal steady state in which there is an ampli-fying wave. As the Laplace contour is pushed upward toward the wr axis in the w plane, one or moreroots, kn, of D(w,kn) = 0 move across the kr axis in the k plane. This is illustrated in Fig. 11.12.8by the superelectric second order system (P > 0 and IMI > 1). Here, for a large both poles, kn, arein the lower half of the k plane. As o is decreased keeping wr constant, one of the poles terminatesin the lower half plane while the other passes through the kr axis into the upper half plane.

Remember that "pushing" the Laplace contour to the wr axis in the w plane is no more than a wayto approximate the inverse Laplace integral in the limit t - c. The function represented by the in-verse Laplace integral must be the same, regardless of the integration contour. This requires thatas the Laplace contour is moved upward, the Fourier integration contour must be distorted so thatwhen it is evaluated by closing the contour, that contour includes the same poles of D(w,kn). Thatis, it must continue to include those that have passed through the kr axis into the upper k plane.This "analytic continuation"3 of the Laplace transformation is therefore obtained by using Fouriercontours revised as illustrated in Fig. 11.12.9. By allowing the Fourier contour, Fig. 11.12.9, tobe distorted so as to enclose the same singularities, the domain of the w plane over which C(w,z) isanalytic has been extended so that the Laplace contour can be that illustrated in Fig. 11.12.6.

For the second order superelectric system, there are now no poles ejclosed by the Fourier con-tour closed in the upper half plane and hence appropriate to evaluating ý(z,w) in the upstream regionz < 0. Thus, the response C(z,t) for z < 0 is zero. Note that this is true at any time and not justfor the asymptotic response. Moreover, for closure in the lower half plane and hence z > Z the

3. P. M. Morse and H. Feshbach, Methods of Theoretical Physics, McGraw-Hill Book Company, 1953,p. 392.

11.51 Sec. 11.12

0 .5 1O=0 -----

(r r

Fig. 11.12.8. Loci of k for loci of w shown by inset. Broken curvesare values of k for real w. Superelectric case (IMI > i,P > 0) displays amplifying waves (P = 1, M = 1.5).

k.,

/'CFI

Fig. 11.12.9

As Laplace contour is pushed tokr w, axis. Dole k,+ crosses k_. axis.

indicating wave amplification.

ýýXk1

summation for the Laplace transform is over both spatial modes. That is, Eq. 14 becomes

-jk z

[g(kl)e

-1z- g(k-1 )e(20)

j(W- W )(M 2 - 1)(k_ - k1 )

Thus, in the long-time limit the Laplace integration along the contour shownin

in Fig. 11.12.6 results

j(wot-klz)=(z,t)= _ _ M2 [g(k+l)e

(k- 1 - k+l)(M2 - 1)

j(wt-k1iz)- g(k-1 )e ]

Again, remember that the real expression is obtained from this expression asEq. 5.

described following

What has been obtained for the second order example is the same sinusoidal steady-state solutionsummerized by Eq. 11.11.21c. If the driving frequency is less than c, one of the downstream wavesdecays away from the source while the other amplifies.

Sec. 11.12.

((z,w) =

(21)

~

wai, oek coss xs

11.52

Criterion Based on Mapping Complex k as a Function of Complex w: Causality requires that the

Laplace contour remain below all singularities of E(w,z). In the process of pushing the Laplace con-tour upward in the w plane to discover the asymptotic response, singularities of D'(w,kn) must beidentified as possible singularities of the Laplace transform, Eq. 14. In the second order system,it is possible to solve explicitly to determine the frequencies, ws, at which

3DD'(ws,k )k [- (Ws,k)]k=k = 0 (22)

n

In general, these possible singularities are more difficult to identify. However, they can be deter-mined by examining the dispersion equation.

Remember that Eq. 22 is really an expression for ws, because by definition

D(ws,k s ) = 0 (23)

In the neighborhood of (Ws,ks), w and k are related by

D D+ 1 2D 2D(w,k) = 0 = D(ws,ks ) + )+ +k (k-ks) + 2 (k-k ) + "' (24)

s ss k s k s 2 k2 s k s

In view of Eqs. 22 and 23, Eq. 24 approximates the dispersion equation in the neighborhood of(Ws,ks) as

D(w,k) D (132 D) (k - ks)2 = 0 (25)DWks W5 s 2 2 s

S,k 3k W,k

This expression makes evident what is happening in the k plane as w approaches w in the w plane,for Eq. 27 is equivalent to the expression

-2 Ws(W-Ws)

k-k- = + (26)s- 2

SD

3k2 k S

It is concluded that the coalescence of a pair of poles in the k plane is the result of having thefrequency w - ws .

Candidates for poles of (w,z) in the w plane can be identified by mapping loci of the rootsto the dispersion equation in the k plane resulting from varying w = wr - jo to cover the lower halfw plane. This is conveniently done by holding wr at fixed values and decreasing a from o to zero.

In retrospect, for the submagnetic and superelectric second order systems, the coalescenceof roots k and k does not occur in the lower half k plane. These mappings were illustrated byFigs. 11.1l17 and 11.12.8, respectively. But, consider the supermagnetic second order system (1M1>1,P<0). The map, illustrated in this case by Fig. 11.12.10, shows that at w = -j/-P(M2-1) - -joc (onthe wi axis) there is a coalescence of the roots of k, and hence a singularity in the terms of Eq. 14comprising the Laplace transform. However, because both roots are below the Fourier contour, theyboth contribute to the Laplace transform for k < z. In fact, as the roots coalesce, the pair of con-tributions to the Laplace transform are together not singular. That is, the denominator of the in-dividual terms can be evaluated by taking the derivative of Eq. 25 and then using Eq. 26 to substi-tute for k - ks,

3D 2D (kD D (27k k k)=+ -2 2 (-s) (27)

3k 3kk ,k k ,s k ,ss

S S S S S S

where the upper and lower signs refer to the respective roots. Thus, the pair of terms resultingfrom the residues for the respective roots of k have opposite signs and, in the limit, equal mag-nitudes. Rather than being singular, the pair tend to the form 0/0, and can be shown to remainfinite. It follows that the Laplace contour can be pushed to just above the wr axis (Fig. 11.12.6)

Sec. 11.1211.53

Fig. 11.12.10

Loci of k for loci of w shown by in-set. Supermagnetic case (IMI > 1,P < 0) displays ordinary waves, bothpropagating downstream (M = 1.5,P = -1).

to evaluate the asymptotic response as before. In the upper half k plane, there are no roots of k forw on the Laplace contour and hence the response is zero for z < 0. For Z < z, closure of the Fourierintegral in the lower half k plane gives contributions from both k+l and k_1 , so the Laplace trans-form, Eq. 14, has two terms. However, as already pointed out, the only singularity is at w = w0, andso integration on the Laplace contour leads to Eq. 21. Now, k+1 can only be real and the downstreamresponse takes the form of beating traveling waves. This is the same result as given by Eq. 11.11.15and illustrated by Fig. 11.11.4.

In summary, the key to understanding the physical significance of a wave having complex valuesof k for real w is a map of the loci of k that result from varying a from m to 0 for all values ofwr . If the loci of a given root terminate in complex k without crossing the kr axis, then it repre-sents an evanescent wave. That is, in the sinusoidal steady state, the complex k represents a wavethat decays away from the source. On the other hand, if loci cross the kr axis, the mode representsan amplifying wave. At a point where a locus crosses the real k axis, k is obviously real and a isstill positive. Thus, for a crossing of the kr axis, the dispersion equation must display "unstable"values of w for real values of k. It is concluded that a necessary condition for existence of anamplifying wave is that wavelengths exist for which a temporal mode is unstable. That is, for k realthere must be roots of D(w,k) for which mi < 0. As a type of instability that grows spatially ratherthan temporally, the amplifying wave is also termed a convective instability.

11.13 Distinguishing Absolute from Convective Instabilities

In Sec. 11.12, examples were purposely considered which were absolutely stable. Thus, theasymptotic response was in the sinusoidal steady state. A necessary condition for an amplifying wavewas found to be "unstable" values of w given by the dispersion equation for real values of k. How isthis response, which is bounded at a given location, z, to be distinguished from one that grows withtime at a given location?

Criterion Based on Mapping Complex k as Function of Complex m: Suppose that a mapping of thedispersion equation, showing loci of k in the complex k plane resulting from varying a from - to 0in the complex w = wr-ja plane, results in a double pole of the type illustrated in Fig. 11.13.1.Here, as w - w., a pair of roots coalesce, one from the upper half plane and one from the lower halfplane. When it is closed, the Fourier contour can no longer be distorted to include the same poles.Before a reaches zero, CF is caught between the coalescing poles, one coming from above and the othercoming from below.

As for the supermagnetic cast from Sec. 11.12, coalescence of the roots, kn, once again indic-ates the possibility of a pole of F(w,z) in the lower half plane. This time the pole in fact exists,because the roots, kn, are on opposite sides of the Fourier contour. Thus, pushed upward so thatmost of it is just above the wr axis, the Laplace contour is as shown in Fig. 11.13.2. The Fouriercontour is caught between the coalescing poles, always including one or the other when closed in theupper or lower half k plane. Integration along the Laplace contour on the segments CL just above

Secs. 11.12 & 11.13 11.54

branch pole

Fig. 11.13.1. Mapping of loci of k in the k Fig. 11.13.2. Laplace contour forplane for trajectory of w passing through evaluating response as t-*osingularity in Laplace transform, showing with branch pole in lowercoalescence of roots from upper and lower half plane.half plane indicating absolute instability.

the Wr axis gives no contribution as t -+ -. However, integration along the segments, CE, adjacentto the branch-cut and around the pole do give contributions. (Note that the segments to either sideof the branch-cut do not cancel, because the integrand is of opposite sign on one side from theother.) In the long-time limit, the Laplace integration reduces to

j9(kn) j(dt-knz) delim E(z,t) = lim + •( •)ED'(k) e (-z) (1)

t-- t> n D'(,k n 21T

On the contour segment CE, w = ws - ja where a is positive, so the asymptotic response is dominatedby a part that grows with time at a fixed location, z.

It is concluded that if the mapping of the dispersion equation from the lower half w plane intothe k plane discloses a pair of coalescing roots, kn, with one root from the upper half plane and onefrom the lower half plane, then the pair represent an absolute instability. This is sometimes alsocalled a non-convective instability.

Note that if roots, kn, of D(w,kn) are to come from the lower and upper half plane and coalesce,one of them must cross the kr axis. As it does so, a is positive. Hence, a necessary condition forabsolute instability as well as convective instability is that there be "unstable" frequencies forreal values of k. That is, for a wave to be a candidate for either spatial or temporal growth, itmust have temporal modes that are unstable.

One consequence of this observation relates to numerous situations that can be envisioned asconfigurations from the sections and problems of Chap. 8 set into a state of uniform motion. Forexample, consider the jet of liquid described in Sec. 8.15, but now having a stationary equilibriumin which the jet streams in the z direction with velocity U. The dispersion equation is Eq. 8.15.25with w + w - kU and takes the form

(w - kU)2 = f(k) (2)

Convection or not, there are only two temporal modes and only one of these can be unstable. That is,for real values of k, at most one of the two roots, w, can have a negative imaginary part.

If the dispersion equation were to be solved for the real frequency spatial modes, there wouldbe an infinite number, about half appearing to "grow" in the z direction. What is clear from theabove result is that only one of these is a candidate for being an amplifying wave. The others areevanescent waves.

Second Order Complex Waves: The subelectric second order system (IMI < 1 and P > 0, Fig. 11.11.1)exemplifies an absolute instability. The w + k mapping is shown in Fig. 11.13.3. In this case, themapping indicates a branch pole at w = -mc H -j/P(l - M2), as can also be seen directly fromEq. 11.12.16. The contour to be used for inverting the Laplace transform as t + o is indicated inFig. 11.13.4.

In more complicated situations, the mapping is carried out numerically by having a routine fordetermining the roots, kn, of D(o,kn) given the complex frequency w = wr - ja. The pattern of theloci in the neighborhood of the coalescing roots, typified by Fig. 11.13.3, can be used to disclosethe coalescing roots.

Secs. 11.12 & 11.1311.55

Wr= .I

-. I 0 .1

w.=O -1

Fig. 11.13.3. Loci of k for loci of W shown by inset. Subelectric case(IMI < 1, P > 0) displays absolute instability (M = 0.5, P = 1).

Fig. 11.13.4

Laplace contour, CF, appropriate for evaluation

of asymptotic response of subelectric (IMI < 1,

P > 0) second order system. Branch cut of

(w - jos)(W + js) is defined consistent with

branch line for A exp(jO) shown by inset.

11.14 Kelvin-Helmholtz Type Instability

Perhaps the most often discussed instability resulting from the interaction of streams is mod-eled by contacting layers of inviscid fluid, one having a uniform velocity relative to the other.1

In Fig. 11.14.1, the lower layer is initially static while the upper one streams in the z directionwith a uniform z-directed velocity, U. That the interface is subject to what is sometimes also calledBernoulli instability suggests why there is a critical velocity above which the stationary equilibriumis unstable. An intuitive picture makes clear the mechanism. For interfacial deformations that havea long wavelength compared to a or b, mass conservation requires that

vz (a - Q = Ua (1)

1. G. K. Batchelor, Introduction to Fluid Mechanics, Cambridge University Press, London, 1967,pp. 511-517.

Secs. 11.13 & 11.14

S --

11.56

_ •. .aPa

Z. • . _ .

Fig. 11.14.1

In stationary state, lower fluid is staticwhile upper one streams to the right withuniform velocity U. At the interface thereis a discontinuity in fluid velocity, andhence a vorticity sheet.

Suppose that an upward directed external surface force density, T, could be applied to a section ofthe interface. At that point on the interface, normal stress equilibrium for the control volume ofFig. 11.14.2 requires that

d eT=p -p

where, for now, surface tension is ignored. With T applied slowly enough that the effect of the inter-facial deformation on the flow can be pictured as a sequence of stationary states, the steady-stateform of Bernoulli's equation is applicable to the regions above and below:

P + Ppv + pgx = H.; x > 0

P + pbgx = 1Ib; x< 0

Initially, T = 0 and the interface is flat, so substitution of Eqs. 3 into Eq. 2 evaluates Ha - Xb =PaU2 /2. Thus, the difference of Eqs. 3 becomes

d e 1 (U2 2p - P =2 a(U -) + b - Pa

Spd.

*. . ·. · ·

Fig. 11.14.2

Where the interface moves upward the streamingvelocity increases. Thus, the pressure abovedrops and the interface is encouraged to furtherdeform.

The combination of this expression of Bernoulli's equation with mass conservation, Eq. 1, and inter-facial stress equilibrium, Eq. 2, gives an expression for the dependence of the surface deflectionon the externally applied surface force density:

1 2_ 1T = W pp U-[1 r ] + g(P, -

a U 2 a

a

What is on the right is an equivalent surface force density representing the combined effects of thefluid inertia and gravity. (Rayleigh-Taylor instability is not of interest here, so it is assumedthat the heavier fluid is on the bottom, pb-Pa>O.) If positive, it acts downward. The equilibriumis stable if a positive upward deformation results in a positive (restoring) surface force density.Thus, for instability,

AT g(pt - Pa)a 2

ar Pa

The effects of surface tension, finite wavelength and unsteady flow left out of this intuitivepicture are brought in by a small-amplitude model that is little different from that developed inSec. 8.9. What has changed is the transfer relation for the streaming layer, which in view ofEq. (c) from Table 7.9.1, is

Sec. 11.14

~U U

- (C = 0) > 0 =# < U

11.57

p k_ -coth ka sinh ka x

= k (7)

ýd -1 -^dp -1 coth kasinh ka x

and the linearized relation between the vertical fluid velocity at the interface and the surface de-formation (Eq. 7.5.5):

^d ^ev = v =X j( - kzU) (8)x x

These expressions replace Eqs. 8.9.4a and 8.9.6 in an otherwise unchanged derivation resulting in thedispersion equation

(w - kzU)2 a coth ka 22 b coth k 22k + k = k 2 + g(b a (9)

This expression is quadratic in w but transcendental in k. Thus, there are only two temporal modes,but an infinite number of spatial modes. In the limit U + 0, these are discussed in Sec. 8.9.

Consider first the temporal modes, having frequencies that follow from Eq. 9:

kzUpka -coth ka coth kb= (10)

pa coth ka + pb coth kb

It follows that the temporal mode exhibits an oscillatory instability if the U is large enough to makethe radicand negative. For instability,

u2 tanh kb +tanh ka k3 + (p - P kU> a k2 b a (11)

Deformations with wavenumbers in the direction of streaming are most unstable, so in the following,k = kz . The first wavelength to become unstable as U is raised can be found analytically in twolimits. First, in the short-wave limit, where ka and kb >> 1, Eq. 11 reduces to

U2 > + i Yk + b - (12)b ]a k k

This expression exhibits a minimum at the Taylor wavenumber

k b (13)

which is familiar from Secs. 8.9 and 8.10. Again, this is the wavenumber of incipient instability andEq. 12 gives the critical velocity as

[4g(P- Pa) Y]aI 1 (14)

In the opposite extreme of long waves (ka and kb << 1) Eq. 11 becomes

U2 b> + [yk2 b a) (15)

In this limit, the first wavenumber to become unstable as U is raised is zero and the critical veloc-

ity is

Sec. 11.14 11.58

U = + b - p) (16)

'b a

Note that, in the limit b/pb << a/Pa, this condition is the same as obtained by the intuitive argu-ments leading to Eq. 6.

With the understanding that the temporal modes so far considered here represent the response toinitial conditions that are spatially periodic, or the response of a system that is reentrant in thez direction, note that the salient difference between the Kelvin-Helmholtz instability and theRayleigh-Taylor instability of Sec. 8.9 is that the former appears oscillatory under the critical con-dition and as a growing oscillation for conditions beyond incipience. This is by contrast with theRayleigh-Taylor instability which is static at incipience and grows exponentially. Note that the dy-namic state at incipience is not included in the arguments leading to Eq. 6.

In connection with instabilities of the Rayleigh-Taylor type, it can be argued that because in-cipience occurs with a vanishing time rate of change, effects of fluid viscosity are ignorable. Here,the dynamic nature of the incipience points to inadequacies of the inviscid model.

To demonstrate the instability without there being more dominant mechanisms of instabilitycoming into play, the configuration shown in Fig. 11.14.1 is arranged so that the upper fluid entersfrom a plenum to the left through a section that is perhaps of a length several times the channelheight, a, to establish the essentially uniform velocity profile. This distance cannot be too large,or viscous effects from the wall and interface will have expanded into the mainstream. What is notdesired is anything approaching a fully developed flow for z > 0. In fact, the viscous boundarylayer will continue to expand over the interface, and what has been described is deformations of theinterface that have effective lengths large compared to the dimensions of the boundary layer.

Complications that are not accounted for by the model include the following. First, if the in-terface is clean, the lower fluid will be set into motion by the viscous shear stress from thestreaming fluid. Thus, the postulated static fluid equilibrium for the fluid below is not strictlyvalid. One way to avoid this complication would be to "turn on" the flow abruptly and establish theinstability before there is time for much effect of the viscous shear stress. Another is to have an.interface supporting a monomolecular film.1 This would compress in the z direction, until the inter-face would be brought to a static state by the gradient in tension of the film. For certain typesof films the incremental dynamics from this static equilibrium would then be as described here, witha surface tension consistent with incremental deformations about this static equilibrium.

What is actually observed at the interface could also be unrelated to the Kelvin-Helmholtz in-stability as modeled here because the developing boundary layer has become unstable over the inter-facial region being observed. Boundary layer instability results in growing perturbations having ascale on the order of the boundary layer thickness.

Certainly, if the fluid entering from the left has a fluctuating component to begin with, inter-facial motions would result that had little to do with the model. These are the types of difficultiesthat often make ambiguous association of the Kelvin-Helmholtz model with effects of wind and water.

In using the Kelvin-Helmholtz model to explain and predict phenomena, it is important to knowwhether it predicts absolute or convective instability. Does the interfacial deflection at a givenposition in Fig. 11..14.1 tend to grow in amplitude until it reaches a state of saturation, or is itcapable of responding to an upstream sinusoidal excitation as a spatially growing wave?

Of the infinite set of spatial modes, only one exhibits a crossing of the kr axis, and then onlyif Eq. 11 is satisfied. Thus, there is only one candidate for an amplifying wave. The other spatialmodes must be evanescent, and are present to satisfy longitudinal boundary conditions.

Consider the (first four) lowest order spatial modes in the long-wave limit (ka and kb << 1).The dispersion equation, Eq. 9, is then quartic in k and quadratic in w:

k4 + k2[G - U2 ] + rUwk - w2 = 0 (17)

where the wavenumber is normalized to an arbitrary length, Z, so that

k = k/k; w = w (18)

1. J. T. Davis and E. K. Rideal, Interfacial Phenomena, Academic Press, 1963, Chap. 5.

Sec. 11.1411.59

2-g(Pb - Pa) apb

= ; =rE 2/ 1 +ay ba (18 cont.)

- ay

The loci of the four roots to this expression in the complex k plane obtained by letting wi in-crease from -o to 0 at fixed values of yr is shown in Fig. 11.14.3. It is clear that for the param-eters summarized in the caption, there is a coalescence of roots coming from the lower and upper halfplanes. Thus, it is concluded that the instability is absolute and grows with time at a fixed loca-tion. This same conclusion is reached by Scott2 in the short-wave limit (Ikal and Ikb >>1).

Fig. 11.14.3

For Kelvin-Helmholtz instability, loci ofcomplex k resulting from varying w as shownby inset. Coalescence of poles from upperand lower planes indicates that for param-eters chosen, instability is absolute(G = 1, U = 2, y = 0.1).

Spatially growing waves associated with wind blowing over water appear to require a model withmore physical content than the simple one considered here.

11.15 Two-Stream Field-Coupled Interactions

Electric or magnetic coupling between streams having different velocities can occur without thenecessity for physical contact between the streams. Thus, the long-wave models developed in Sec. 11.10(Fig. 11.10.4) make a more satisfying representation of streaming interactions than does the traditionalKelvin-Helmholtz model of Sec. 11.14.1 Transverse deflections of the respective streams are describedby Eqs. 11.10.19-11.10.22, where e and v are as defined by Eq. 11.10.4. With fl and f2 linearized,these expressions become

2+ 32 1 1

(- + M_ ) = + Pr P(13t 1 zz 1 2 1- P2

a a 2 2 1(- + M2 t 2 2) 2 2 - 2 1)

3z

Normalization is as given by Eqs. 11.10.3 and 11.10.6. Remember, what is described is a pair of"strings" respectively convecting in the z direction with Mach numbers M1 and M 2 . With an electricfield coupling the streams, P is positive, whereas with a magnetic field, P is negative. The "super"and "sub" responses to initial conditions and boundary conditions consistent with causality are dis-cussed in detail in Sec. 11.10. There, the method of characteristics is used to describe the dynamicsin real space and time.

The discussion is now limited to a description of the interaction of a stream with a fixed"string." That is, M2 = 0 and M1 E M. The model is typical of interactions between electron beams

2. A. Scott, Active and Nonlinear Wave Propagation in Electronics, Wiley-Interscience, New York,1970, pp. 83-87.

1. For a detailed discussion of the many possibilities for the model considered in this section, seeF. D. Ketterer and J. R. Melcher, "Electromechanical Costreaming and Counterstreaming Instabili-ties," Phys. Fluids 11, 2179-2191 (1968) and "Electromechanical Stream-Structure Instabilities,"Phys. Fluids 12, 109-117 (1969).

Secs. 11.14 & 11.15 11.60

and transmission lines and plasmas. Counterstreaming and costreaming cases are considered in heproblems.

What are the conditions for instability of the temporal modes, and is a given mode absolut ly orconvectively unstable?

Substitution of complex amplitudes in Eqs. 1 and 2 results in the dispersion equation

[( - Mk)2 k2 + ][2 _ k2 1 p2 0 (3)

This expression is quartic in either w or k. Subroutines for numerically finding the roots of poly-nomials are readily available.

Consider first the superelectric case, where M > 1 and P > 0. The temporal modes are summerizeaby the plots of complex w for real k illustrated in Fig. 11.15.1. For M > 2 and M < 2, respectively,Parts (a) and (c) illustrate the dynamics of the stream if the structure were held fixed, and of thestructure if the stream were fixed. These are the superimposed dispersion relations for an absoluteand a convective instability.

The self-consistent solutions to Eq. 3 are illustrated by Parts (b) and (d). For M > 2, Pt. (b),very small k results in four roots having the same real part and forming two complex conjugate pairs.At a midrange of k, two complex conjugate pairs result with the pairs having different real parts butimaginary parts of the same magnitude. At high k, waves are formed that are typical of those on theconvecting string and fixed string respectively.

Fig. 11.15.1

Complex w as a function of real k for

electric stream structure (P > 0 r-

action. Parts (a) and (c), which aredrawn with mutual coupling terms ignored,are shown for respective comparison toParts (b) and (d) (P = 1).

For 1 < M < 2, an overstability results over a range of k where the uncoupled systems show noinstability. This has a character that would not be expected from either the stream or the structurealone.

The plots of complex k for fixed wr as Wi is increased from -o to 0, shown in Fig. 11.15.2, dis-

close the character of these instabilities. One pair of roots move from above and below the kr axisto coalesce in the upper half plane. Thus, this pair represent an absolute instability. The otherpair migrate upward from the lower half plane, one extending into the upper half plane. This repre-

sents an amplifying wave.

It might be expected that the absolute instability eventually dominates the response. In the

next section, a physical demonstration illustrates how this same system, confined to a finite length,

can exhibit an oscillatory overstability that can be traced to the wave amplification.

Sec. 11.1511.61

Fig. 11.15.2

For superelectric stream inter-acting with fixed "string"(P > 0) mapping of complex kplotted for fixed wr as wi isincreased from -o to 0.

Fig. 11.15.3

(a) (b) Complex w as a function of real k formagnetic stream-structure (P < 0) in-teraction. Parts (a) and (c), whichare drawn with mutual coupling termsignored, are shown for respective com-parison to Parts (b) and (d) (P = -1).

(c) (d)

Consider next the supermagnetic case, where M > 1 and P < 0. Now, the temporal modes are as

shown by the dispersion plots of Fig. 11.15.3. Purely propagating waves result if 1 < M < 2. How-

ever, if M is in the range 2 < M, there is a range of k over which overstability is exhibited. Be-

cause the uncoupled stream and structure are completely stable, this temporal mode instability has

some of the character of the Kelvin-Helmholtz instability. Note that it is similar to the high wave-

number overstability exhibited for the superelectric case by Fig. 11.15.1d. Whether the force on one

stream due to a deflection of the other pushes or pulls, the result is a growing oscillation.

By contrast with the Kelvin-Helmholtz instability, this one is convective. This follows

from the mapping of complex k for complex w in the lower half plane illustrated (for this 2 < M case)

by Fig. 11.15.4. The instability is indeed convective. The example is also worthy of remembrance

because it illustrates how longitudinal boundary conditions, which have not yet come into play here,

can have a dramatic effect. In Sec. 11.16, it is found that with boundary conditions (consistent

with causality) this convective instability can become absolute.

Sec. 11.15 11.62

Fig. 11.15.4

For supermagnetic stream inter-acting with fixed "string"(P < 0), mapping of complex kplotted for fixed wr as Wi isincreased from -- to 0.

11.16 Longitudinal Boundary Conditions and Absolute Instability

Quasi-one-dimensional linearized models, which retain the fundamental modes of a two-dimensionalsystem having an infinite set of modes, can be obtained from the exact model by taking the long-wavelimit. The coupled second order system used as a prototype model in Sec. 11.15 is one example. Thefollowing steps are generally applicable to determining the eigenfrequencies of such systems subjectto homogeneous longitudinal boundary conditions. From a point of view somewhat different from thattaken in Sec. 11.10, this section also emphasizes the importance of being sure that the boundary con-ditions are consistent with causality.

Suppose that there are N spatial modes. That is, with the frequency specified, there are Nroots, kn, to the dispersion equation D(w,kn) = 0:

k = k (w) (1)n n

Then, the response can be written as

N -jkz

2 = Re( Ane(2)n=1

where An are arbitrary coefficients. Although it remains to be found, the frequency of each term inEq. 2 is the same. Thus, for initial conditions having the z dependence of the function in bracketsin Eq. 2, the response would have the one (generally complex) frequency, w. All other dependent vari-ables can be written in terms of these solutions, and hence in terms of the N coefficients, An.

There are N homogeneous boundary conditions imposed either at z = 0 or at z = Z. Evaluatedusing Eq. 2 and the other dependent variables written in terms of these same coefficients, theseboundary conditions comprise N equations that are linear in the coefficients, An. The condition thatthe coefficient determinant vanish,

Det(w,kl,k2,.--kN) = 0 (3)

together with the dispersion equation, Eq. 1, is the desired eigenfrequency equation.

To be specific, consider the stream coupled (either by the electric field or by a magnetic field)to a stationary continuum. The configuration is as shown physically by Fig. 11.10.4 and described by

Eqs. 11.15.1 and 11.15.2 with M2 = 0. Longitudinal boundary conditions, illustrated schematicallyby Fig. 11.16.1, are

1(0,t) = 0; -z (0,t) = 0; ( 2 (0,t) = 0; ( 2 (Z,t) = 0 (4)

That these are consistent with causality is demonstrated in Sec. 11.10 by appealing to the method of

Secs. 11.15 & 11.1611.63

characteristics. Here, the same conclusions might be reachedby identifying three of the four spatial modes with the up-stream boundary where z = 0 and the other with the downstreamboundary where z = £. Whether the coupling is electric

(Fig. 11.15.2) or magnetic (Fig. 11.15.4), the mapping of thedispersion equation with wi increased from -o to 0 at con-stant wr into the k plane gives rise to three loci originatingin the lower half plane and one in the upper half plane.

The stream deflection is given by Eq. 2 with N = 4. Itfollows from Eq. 11.15.1 that the stationary continuum has adeflection that can be written in terms of these same coef-ficients as

4 -jknz 2 2 2

S= Re( E QnAne )ewt; n -k + P) (5)N=l

The four linear equations obtained by using Eqs. 2 and 5 toexpress the homogeneous boundary conditions, Eqs. 4, give

-j k 19 -jk 2 Z -jk 3 k -jk 4 p.e e e e

Q, Q2 Q3 4

klQ1

k2Q2 k3Q3 k4Q4

U

Fig. 11.16.1. Schematic of stream-structure interaction, show-ing three upstream boundaryconditions and one downstreamcondition for M1>l and M 2=0.

The determinant of the coefficients in this expression set equal to zero takes the form of Eq. 3 and is

a complex equation for the complex variable, w. Remember that the kn's are determined in terms of thefrequency from the dispersion equation found from Eqs. 11.15.1 and 11.15.2.

In general, finding the roots of the complex transcendental combination of Eqs. 3 and 1 is dif-ficult.1 Carrying out a numerical search for the roots is as straightforward as using the root-finding

techniques illustrated by Fig. 5.17.5. However, unless the computer is guided, it can easily fail toconverge on certain roots. One way to obtain solutions in a relatively automated way is to start by

finding the roots in a limit in which one of the parameters is small enough to make an analytical ap-proximation possible. Then, these roots can be followed as that parameter is raised to the desiredvalue.

Numerically determined normalized eigenfrequencies are shown as a function of the normalizedlength in Fig. 11.16.2. Illustrated here is the electrically coupled system (P > 0). The first modereflects the destabilizing effect of the electric field. At low fields, the effect of the coupling is

to produce damping, but as the field is raised, there is a threshold for static instability in thismode, much as if the stationary continuum were coupled to rigid walls. The real part of the frequency

below this threshold and the condition for incipience are little different from what would be obtained

if the stream were rigid.

The effect of the coupling on the second mode is something new. According to the model (which

ignores viscous drag), this mode is overstable with the application of even the slightest electricfield.

In the magnetic case, with eigenfrequencies shown in Fig. 11.16.3, the static instability of the

lowest mode is replaced by a damping that only becomes larger as the field is raised. But, reflecting

a regenerative feedback mechanism, the higher order modes exhibit overstabilities similar to those

for the electric case. The eigenfunctions for the first three modes, illustrated by Fig. 11.16.4, give

some idea as to the origins of this spontaneous oscillation of growing amplitude.

The theoretical second eigenfunction for the electric case, having eigenfrequencies given by

Fig. 11.16.2, has an appearance little different from that of the demonstration experiment shown by

Figs. 11.16.5 and 11.16.6. In the first of these, a time exposure is shown of the water jet coupled

electrically to a stretched spring. A sequence of instantaneous exposures of the same system following

the dynamics through one oscillation is shown in the second figure. Deflections that amplify spatially

on the jet are fed back upstream by the spring with a phase that makes the feedback loop regenerative.

1. F. D. Ketterer, Electromechanical Streaming Interactions, Ph.D. Thesis, Department of Electrical

Engineering, Massachusetts Institute of Technology, 1965.

Sec. 11.16 11.64

That the system with magnetic field coupling displayed the same instability makes it clear that thismechanism of instability has little to do with the nature of the coupling between stream and struc-ture. It is the reflection of a wave on the spring by the downstream boundary condition that turns

the magnetic field configuration from an infinite system exhibiting an amplifying wave into one thatis finite and absolutely unstable.

Fig. 11.16.2

Complex eigenfrequency as a functionof normalized length for superelec-tric stream-structure interaction.

FP/UT -

I-40a

ULu0

z

Fig. 11.16.3

Complex eigenfrequency versus normal-ized length for a magnetic stream-structure system for the lowest threemodes.

a/fPur

Sec. 11.1611.65

O 0.25 0.50 0.75 1.00 125 1.50

NORMALIZED DISTANCE eF,/UT = 1.75

Fig. 11.16.4. Eigenfunctionsfor the three lowest modesfor the magnetic system ofFig. 11.16.3.

1.75

Sec. 11.16

w ±-+1.14

34

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11.17 Resistive-Wall Electron Beam Amplification

Either as a limitation in a linear accelerator or as the basis for making an amplifier,1 theconvective instability induced by the interaction of an electron beam with charges induced on aneighboring wall provides an interesting example of how lossy and propagating systems can interactto produce a spatially growing wave. The planar version shown in Fig. 11.17.1 incorporates the beamdescribed in Sec. 11.5. What has been added is walls if finite conductivity, 0, themselves backedby much more highly conducting material.

With the objective of obtaining the dispersionequation for the beam coupled to the resistive layer,observe that fields in the layer (described inSec. 5.10) are represented by the flux-potentialtransfer relations [Eq. (a) of Table 2.16.1]. Inview of the highly conducting material bounding theresistive layer, Da = 0, and it follows that on thelower surface of the layer,

^b ^bD = sk coth kd 4b

x

At this surface, the potential is continuous

ab c= c

and conservation of charge requires that

a ̂ b(jw + -)D - jWD = 0E X x

In view of Eqs. 1 and 2, this expression becomes onerepresenting the layer as "seen" by the electron beam:

.................................................................

l , C

. db e

g..

....... ..

Fig. 11.17.1. Cross section of planarelectron beam coupled to resis-tive wall.

D =- (jw +) (k coth kd)x jW E

The beam is represented at this same surface by Eq. 11.5.11 (with 6 = 6o) . (Remember that it hasalready been assumed in Sec. 11.5 that electron motions are even.) Thus, the dispersion equation isobtained by setting Dc as given by Eq. 4 equal to Dc as given by Eq. 11.5.11:x x

1 a--1 (jw + --)(Ek coth kd) =jW E

where

2 = k2 ( 1

-E k(k + y coth ka tanh yb)

k coth ka + y tanh yb

2

(w - kU)2

The discussion of temporal and spatial modes for the beam coupled to an equipotential wall given inSec. 11.5 makes it clear that there are an infinite number of either temporal or spatial modes. Be-cause of the Laplacian character of the field that couples the wall to the beam, the interactionbetween wall and beam tends to be strongest for the longest wavelengths. Thus, the long-wave limitof Eq. 5 is now taken by considering ka << 1, kb << 1, and kd << 1. Thus, Eqs. 5 and 6 become apolynomial dispersion equation that is cubic in o and quartic in k,

(jwR. + 1) 1( - k)2 + k 2 -

2 k 2 1 = -jwR rk 2 [(w - k)2(1 + b) ba12e a p e a a p

and the higher order transverse spatial modes are neglected. Here the frequency and wavenumber have.been normalized and dimensionless parameters introduced:

dEb _ Us _ o= U ; R = -S-; r -bERU -e

bwk = kb; w P-- -p U

1. C. K. Birdsall, G. R. Brewer and A. V. Halff, "The Resistive-Wall Amplifier," Proc. IRE 41,865-875 (1953).

L

[

Sec. 11.17 11.68

The modes that have been retained by this long-wave model can be identified by considering two limits

of Eq. 7. In the first, 0 - m (Re + 0) and the expression reduces to one for space-charge oscilla-

tions superimposed on the convection:

wkw = kU+ P

- I+ k2rab

These are the lowest modes from Sec. 11.5. In the second, the beam is removed by setting wp + 0.Then, Eq. 7 requires that

.2[ d1e (1 + b)k2jw = 1+aE bEF 2

(10)

This is the damped charge relaxation mode resulting from the ohmic loss in the layer and energy stor-age both within the layer and in the region of the gap and beam.

What can be expected for the coupled modes? First, to consider the temporal modes, Eq. 7 iswritten as a cubic in w:

3_ b 121+br + ýj 2bb2}2 2

(11[

+w jRk +k(l+r+ ) 2(1+ rb)]- 2k(- + k) +k k(2 _ w2)= 0e a a p a a a p

Numerical solutionfunction of real k.frequencies, and so

of this expression is illustrated by Fig. 11.17.2, where complex WFor the given parameters, the growth rate, W2i, is small compared

it is shown on a separate plot.

is shown as ato the other

Fig. 11.17.2

Plot of dispersion equation for long-wave coupling of electron beam toresistive wall, showing normalizedcomplex w for normalized real k(b/a = 1, r = 1, Re = 1, w = 1).

e --p

Sec. 11.17

S

11.69

That the system is unstable raises the second question. Would this instability result in a beamoscillation that grew in time at a fixed location, or would it result in a spatial amplification ofany disturbance? The limitation on a particle accelerator, or the possibility of using the interac-tion to make an amplifier, hinge on whether the instability is convective or absolute.

In order to plot loci of complex k as w is varied at constant Wr from mi = -w to 0, Eq. 7 isexpressed as a polynomial in k:

k4jR [1 + r(l + )] +1 + k3{2jiW2R [1 + r(1 + b 2f e a e a

+ k2 jR [m + - + r (2 + -b ] + ( + - W-) (12)e a p a a p, a p

+k -2w -(jwR + 1) + (jwR + 1) = 0a e a e

The loci of the four roots to Eq. 12 obtained by varying wi from -- to 0 with Wr fixed are shownin Fig. 11.17.3. Of course, all values of wr must be considered. The ones shown are typical. Ap-parently the instability is convective. Thus, fluctuations on the beam as it enters the interactionregion would amplify in space. To contend with an undesired instability on a beam having a givenentrance fluctuation level, the length of the interaction region would have to be limited.

Fig. 11.17.3

For resistive wall interactingwith electron beam, the loci ofthe four long-wave modes in thek plane are shown as wi is variedfrom - to 0 holding 1r fixed.The inset shows the associatedtrajectories of w in the complex1 plane. The instability isapparently convective.

The problems give some hint of the wide variety of physical situations in which amplifying wavesresult from coupling between charged particle streams and various types of structures and media.Electron beam interactions and plasma dynamics are rich with examples of the various forms of complexwaves. So also is fluid mechanics, where boundary layer instabilities and other precursors of turbu-lence are often amplifying waves.

Sec. 11.17

_

11.70